510:
6434:
1758:
417:; based on that principle, as light travels through slits and boundaries, secondary point light sources are created near or along these obstacles, and the resulting diffraction pattern is going to be the intensity profile based on the collective interference of all these light sources that have different optical paths. In the quantum formalism, that is similar to considering the limited regions around the slits and boundaries from which photons are more likely to originate, and calculating the probability distribution (that is proportional to the resulting intensity of classical formalism).
713:
522:
4874:
761:
6446:
671:
566:
338:
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649:
745:
90:
5247:
take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.
629:
1766:
4444:, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger diameter, and hence a lower divergence. Divergence of a laser beam may be reduced below the diffraction of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity. This may result in a
498:
554:
40:
241:
6482:
679:
1985:
467:, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes, we will have to take into account the full three-dimensional nature of the problem.
613:
3066:
3061:
6458:
6494:
4066:
3627:
6470:
4403:
4451:
When the wave front of the emitted beam has perturbations, only the transverse coherence length (where the wave front perturbation is less than 1/4 of the wavelength) should be considered as a
Gaussian beam diameter when determining the divergence of the laser beam. If the transverse coherence length
400:
for the photon: the light and dark bands are the areas where the photons are more or less likely to be detected. The wavefunction is determined by the physical surroundings such as slit geometry, screen distance, and initial conditions when the photon is created. The wave nature of individual photons
840:
Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire
5246:
The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not
1996:
776:
An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit interference effects can be calculated. The analysis of
5261:
If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double-slit experiment, this
4885:
The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the
459:
It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and, in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another
4725:
Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The
3830:
808:
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to
3282:
2785:
5455: : It has illuminated for us another, fourth way, which we now make known and call "diffraction" , because we sometimes observe light break up; that is, that parts of the compound , separated by division, advance farther through the medium but in different , as we will soon show.
4782:
is a useful theorem stating that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape, but with differing intensities. This means that the interference conditions of a single obstruction would be the same as that of a single slit.
5254:. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an
4764:
is another diffraction phenomenon. It is a result of the superposition of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly.
3409:
2845:
5447: : Nobis alius quartus modus illuxit, quem nunc proponimus, vocamusque; diffractionem, quia advertimus lumen aliquando diffringi, hoc est partes eius multiplici dissectione separatas per idem tamen medium in diversa ulterius procedere, eo modo, quem mox declarabimus.
1785:
3417:
3835:
5282:. These femtosecond-duration pulses will allow for the (potential) imaging of single biological macromolecules. Due to these short pulses, radiation damage can be outrun, and diffraction patterns of single biological macromolecules will be able to be obtained.
4822:
using a plane wave spectrum formulation. A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by
1226:
4491:. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane whose radius (as measured to the first null) is
5694:(Proposition 1. Light propagates or spreads not only in a straight line, by refraction, and by reflection, but also by a somewhat different fourth way: by diffraction.) On p. 187, Grimaldi also discusses the interference of light from two sources:
4923:
According to quantum theory every particle exhibits wave properties and can therefore diffract. Diffraction of electrons and neutrons is one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the
1750:
5019:
has been observed for small particles, like electrons, neutrons, atoms, and even large molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids, small molecules and proteins.
4182:
1801:
A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles
2126:
3707:
2481:
4858:
5563:
Juffmann, Thomas; Milic, Adriana; MĂĽllneritsch, Michael; Asenbaum, Peter; Tsukernik, Alexander; TĂĽxen, Jens; Mayor, Marcel; Cheshnovsky, Ori; Arndt, Markus (25 March 2012). "Real-time single-molecule imaging of quantum interference".
1969:
The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.
3147:
509:
3702:
768:
A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity, in accordance with the
781:, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by
1890:
4415:
is the spatial
Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see
3154:
2657:
4696:
380:
of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.
227:
The amount of diffraction depends on the size of the gap. Diffraction is greatest when the size of the gap is similar to the wavelength of the wave. In this case, when the waves pass through the gap they become
4730:
specifies that two point sources are considered "resolved" if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.
2355:
3287:
1713:
5999:
1789:
5470:
1792:
1791:
1787:
1786:
804:
or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit.
1793:
4440:
profile and has the lowest divergence for a given diameter. The smaller the output beam, the quicker it diverges. It is possible to reduce the divergence of a laser beam by first expanding it with one
1429:
4877:
The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.
521:
956:
309:
demonstrating interference from two closely spaced slits. Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves.
1138:
541:
The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a
4734:
Thus, the larger the aperture of the lens compared to the wavelength, the finer the resolution of an imaging system. This is one reason astronomical telescopes require large objectives, and why
2597:
4174:
5262:
would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single-slit diffraction patterns.
1102:
5787:
4119:
3056:{\displaystyle \Psi (r)\propto \iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')~{\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,dx'\,dy',}
5857:
Fresnel, Augustin-Jean (1818), "Mémoire sur la diffraction de la lumière" ("Memoir on the diffraction of light"), deposited 29 July 1818, "crowned" 15 March 1819, published in
4896:
When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a
4846:
883:
4962:
5104:
2014:
1790:
473:
4857:
4527:
2837:
1348:
2399:
1381:
3622:{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}
2386:
5012:
of the particle (mass Ă— velocity for slow-moving particles). For example, a sodium atom traveling at about 300 m/s would have a de
Broglie wavelength of about 50 picometres.
4061:{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik\sin \theta (\cos \phi x'+\sin \phi y')}\,dx'\,dy'.}
1003:
910:
841:
slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is approximately
2626:
1743:
1594:
756:
Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.
1560:
5046:) in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.
2531:
2197:
1532:
1480:
2648:
2278:
1917:
1053:
5058:. Bragg diffraction is a consequence of interference between waves reflecting from many different crystal planes. The condition of constructive interference is given by
1255:
836:
5486:
Wireless
Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002 pg 126
1310:
5124:
4547:
1814:
1023:
5265:
In the case of particles like electrons, neutrons, and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.
5205:) whose wavelength is on the order of (or much smaller than) the atomic spacing. The pattern produced gives information of the separations of crystallographic planes
5164:
1503:
135:
4637:
169:, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. If there are multiple,
2256:
2224:
1282:
802:
596:- bright rings around the shadow of the observer. In contrast to the corona, glory requires the particles to be transparent spheres (like fog droplets), since the
5987:
5223:
5184:
5144:
5006:
4982:
4716:
4611:
4591:
4567:
3076:
2166:
2146:
1957:
1937:
1122:
976:
111:
3634:
2601:
This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector
2486:
282:
4650:
4398:{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-i(k_{x}x'+k_{y}y')}\,dx'\,dy',}
485:
372:. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative
3825:{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} ,}
269:, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in
2294:
497:
1603:
254:
might have observed diffraction in a broadening of the shadow. The effects of diffraction of light were first carefully observed and characterized by
1788:
4893:
The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
5941:
in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal
Academy of Sciences.)
4452:
in the vertical direction is higher than in horizontal, the laser beam divergence will be lower in the vertical direction than in the horizontal.
1028:
A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles
6117:
Kouyoumjian, R. G.; Pathak, P. H. (November 1974). "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface".
6409:
6433:
6261:
6234:
6209:
6182:
6032:
5654:
5507:
4467:
5935:(I have thus shown in what manner one can imagine that light propagates successively by spherical waves, … ) (Note: Huygens published his
915:
4803:
that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building. The knife-edge effect is explained by the
5764:
5698:(Proposition 22. Sometimes light, as a result of its transmission, renders dark a body's surface, previously illuminated by another .)
2536:
5924:
5692:"Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte."
5913:
5683:
5434:
4124:
421:
302:
5397:
4900:
pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.
578:
This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the
6157:
5547:
4845:
4828:
1058:
4073:
2230:. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
1757:
5696:"Propositio XXII. Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam."
3277:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}},}
2780:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}
4930:
1387:
401:(as opposed to wave properties only arising from the interactions between multitudes of photons) was implied by a low-intensity
5342:
5065:
166:
4494:
6267:
5228:
For completeness, Bragg diffraction is a limit for a large number of atoms with X-rays or neutrons, and is rarely valid for
4804:
770:
592:- a bright disc and rings around a bright light source like the sun or the moon. At the opposite point one may also observe
357:
146:
5278:
A new way to image single biological particles has emerged since the 2010s, utilising the bright X-rays generated by X-ray
1962:
The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a
674:
View from the end of
Millennium Bridge; Moon rising above the Southwark Bridge. Street lights are reflecting in the Thames.
463:
The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For
5933:"J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques, … "
4836:
4412:
3404:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}
364:. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a
6424:
720:
Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.
5327:
5312:
4807:, which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, and creates a new
712:
6519:
2494:
2393:
698:, that is diffraction off the meat fibers. All these effects are a consequence of the fact that light propagates as a
619:
589:
326:
5050:
Diffraction from a large three-dimensional periodic structure such as many thousands of atoms in a crystal is called
4461:
1434:
724:
667:
in camera or support struts in telescope; In normal vision, diffraction through eyelashes may produce such spikes.
635:
593:
5832:
Fresnel, Augustin-Jean (1816), "Mémoire sur la diffraction de la lumière" ("Memoir on the diffraction of light"),
844:
445:
414:
205:
181:
5258:, the coherence length is related to the lifetime of the excited state from which the atom made its transition.
4873:
690:
which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When
298:
255:
75:
1221:{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}
2792:
6119:
5757:
5297:
4432:
changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially
1317:
1314:
which is also a normalization factor of the intensity profile that can be determined by an integration from
585:
361:
4742:(large aperture diameter compared to working distance) in order to obtain the highest possible resolution.
2360:
1353:
70:
of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the
5352:
5332:
5317:
5193:
Bragg diffraction may be carried out using either electromagnetic radiation of very short wavelength like
4897:
4779:
4774:
1445:
1132:
981:
888:
670:
429:
402:
244:
Thomas Young's sketch of two-slit diffraction for water waves, which he presented to the Royal
Society in
1507:
hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If
5372:
5292:
5194:
4925:
1721:
1572:
1452:
1128:
739:
397:
337:
310:
6445:
565:
2003:
The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the
1539:
760:
6514:
6294:
6061:
5796:
5583:
5392:
5347:
5229:
5202:
4918:
420:
There are various analytical models which allow the diffracted field to be calculated, including the
5836:, vol. 1, pp. 239–81 (March 1816); reprinted as "Deuxième Mémoire…" ("Second Memoir…") in
5468:"A History of Physics in its Elementary Branches, including the evolution of physical laboratories."
5034:
2500:
2171:
1995:
1562:
would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of
645:
A shadow of a solid object, using light from a compact source, shows small fringes near its edges.
5715:
5402:
5279:
5241:
5198:
5055:
4914:
4488:
4433:
4411:
of the aperture distribution. Huygens' principle when applied to an aperture simply says that the
2651:
2604:
2490:
1779:
1597:
1511:
1459:
778:
542:
437:
294:
174:
170:
158:
5739:
5639:
Stationary
Diffraction by Wedges : Method of Automorphic Functions on Complex Characteristics
5467:
2631:
2261:
2121:{\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}
6498:
6486:
6450:
6326:
6281:
Neutze, Richard; Wouts, Remco; van der Spoel, David; Weckert, Edgar; Hajdu, Janos (August 2000).
6087:
5878:
5814:
5615:
5573:
5377:
4832:
4739:
4735:
4727:
4644:
3151:
In the far field, wherein the parallel rays approximation can be employed, the Green's function,
2288:
1895:
1563:
1127:
There is no such simple argument to enable us to find the maxima of the diffraction pattern. The
1031:
660:
441:
433:
322:
197:
193:
5965:
1231:
813:
777:
this system is simplified if we consider light of a single wavelength. If the incident light is
752:
448:. Most configurations cannot be solved analytically, but can yield numerical solutions through
6401:
2489:.) By direct substitution, the solution to this equation can be readily shown to be the scalar
2476:{\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi ).}
1289:
6375:
6318:
6310:
6257:
6230:
6205:
6178:
6153:
6028:
6022:
5937:
5894:
5890:
5874:
5866:
5761:
5650:
5607:
5599:
5543:
5503:
5407:
5322:
5307:
5109:
5051:
5029:
4819:
4532:
4470:
The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this
4408:
4177:
2839:
is incident on the aperture, the field produced by this aperture distribution is given by the
1008:
648:
571:
Data is written on CDs as pits and lands; the pits on the surface act as diffracting elements.
385:
349:
293:) observed the diffraction patterns caused by a bird feather, which was effectively the first
251:
224:
and, therefore, undergoes diffraction (which is measurable at subatomic to molecular levels).
217:
142:
6049:
5841:
5149:
1488:
479:
Computer-generated intensity pattern formed on a screen by diffraction from a square aperture
120:
6462:
6365:
6357:
6302:
6128:
6099:
6069:
5845:
5804:
5782:
5680:
5642:
5591:
5535:
5431:
5251:
4824:
4616:
2840:
2389:
2388:
is the 3-dimensional delta function. The delta function has only radial dependence, so the
2281:
744:
716:
Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry
453:
185:
71:
5921:
2241:
2202:
1260:
784:
5995:
5928:
5917:
5768:
5687:
5474:
5438:
5302:
4985:
4757:
4751:
2227:
687:
410:
318:
314:
306:
270:
245:
83:
5709:
6416:
6298:
6065:
5800:
5733:
5587:
89:
62:
is the interference or bending of waves around the corners of an obstacle or through an
6438:
6370:
6345:
6090:(June 2013). "GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations".
5885:, American Book Company, 1900, pp. 81–144. (First published, as extracts only, in
5362:
5357:
5337:
5208:
5169:
5129:
5039:
4991:
4967:
4719:
4701:
4596:
4576:
4552:
4480:
4417:
2151:
2131:
1942:
1922:
1107:
961:
449:
321:, and thereby gave great support to the wave theory of light that had been advanced by
96:
5633:
Komech, Alexander; Merzon, Anatoli (2019), Komech, Alexander; Merzon, Anatoli (eds.),
1765:
161:
source (such as a laser) encounters a slit/aperture that is comparable in size to its
6508:
5818:
5539:
4761:
4472:
4445:
4437:
2284:
425:
373:
369:
4863:
Diffraction on a soft aperture, with a gradient of conductivity over the image width
4466:
628:
6474:
6330:
5823:. (Note: This lecture was presented before the Royal Society on 24 November 1803.)
5619:
5367:
5255:
406:
393:
365:
274:
5907:
3832:
the expression for the
Fraunhofer region field from a planar aperture now becomes
39:
6251:
5497:
1885:{\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}
6395:
5783:"The Bakerian Lecture: Experiments and calculations relative to physical optics"
5646:
5382:
5016:
4910:
4811:. This new wavefront propagates into the geometric shadow area of the obstacle.
4477:
4441:
1963:
597:
553:
342:
240:
221:
213:
6073:
5250:
The length over which the phase in a beam of light is correlated is called the
656:) seen in the center of the shadow of a circular obstacle is due to diffraction
6171:
5752:
Letter from James
Gregory to John Collins, dated 13 May 1673. Reprinted in:
5634:
5387:
4815:
1749:
1717:
The choice of plus/minus sign depends on the definition of the incident angle
695:
678:
653:
601:
579:
528:
229:
162:
31:
6314:
6103:
5603:
4808:
4800:
4487:
The ability of an imaging system to resolve detail is ultimately limited by
2008:
2004:
1979:
1600:), the intensity profile in the Fraunhofer regime (i.e. far field) becomes:
1441:
723:
Diffraction can also be a concern in some technical applications; it sets a
691:
464:
377:
313:
did more definitive studies and calculations of diffraction, made public in
201:
150:
114:
44:
6379:
6361:
6322:
6282:
6132:
5809:
5738:. London: Longman, Rees, Orme, Brown & Green and John Taylor. pp.
5611:
5428:
Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo
157:. The characteristic bending pattern is most pronounced when a wave from a
5595:
3142:{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} .}
1984:
17:
5009:
4640:
4570:
4176:
the Fraunhofer region field of the planar aperture assumes the form of a
3697:{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }
705:
Diffraction can occur with any kind of wave. Ocean waves diffract around
664:
63:
52:
4881:
Several qualitative observations can be made of diffraction in general:
612:
503:
Computational model of an interference pattern from two-slit diffraction
3065:
154:
82:
and was the first to record accurate observations of the phenomenon in
34:, the change in direction of a wave passing from one medium to another.
6350:
Philosophical Transactions of the Royal Society B: Biological Sciences
5232:
or with solid particles in the size range of less than 50 nanometers.
4691:{\displaystyle \theta \approx \sin \theta =1.22{\frac {\lambda }{D}},}
6306:
1596:
of the light onto the slit is non-zero (which causes a change in the
1005:
is the angle of incidence at which the minimum intensity occurs, and
389:
67:
6344:
Chapman, Henry N.; Caleman, Carl; Timneanu, Nicusor (17 July 2014).
4827:, who introduced the notion of diffraction coefficients through his
6469:
5677:
Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis …
1769:
A diffraction pattern of a 633 nm laser through a grid of 150 slits
515:
Optical diffraction pattern (laser, analogous to X-ray diffraction)
388:
understanding of light propagation through a slit (or slits) every
51:
beam projected onto a plate after passing through a small circular
6283:"Potential for biomolecular imaging with femtosecond X-ray pulses"
5859:
Mémoires de l'Académie Royale des Sciences de l'Institut de France
5578:
5033:
4872:
4429:
3414:
The expression for the far-zone (Fraunhofer region) field becomes
3064:
2350:{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}
1994:
1983:
1783:
1764:
1756:
1748:
1448:), that is, at a distance much larger than the width of the slit.
759:
751:
743:
711:
706:
677:
669:
647:
545:
to form the familiar rainbow pattern seen when looking at a disc.
336:
263:
239:
209:
189:
165:, as shown in the inserted image. This is due to the addition, or
88:
48:
38:
1999:
Diffraction pattern from a circular aperture at various distances
5920:(Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From
4887:
1708:{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left}
699:
353:
290:
286:
5883:
The Wave Theory of Light: Memoirs by Huygens, Young and Fresnel
5054:. It is similar to what occurs when waves are scattered from a
4407:
In the far-field / Fraunhofer region, this becomes the spatial
491:
Generation of an interference pattern from two-slit diffraction
3073:
where the source point in the aperture is given by the vector
5754:
Correspondence of Scientific Men of the Seventeenth Century …
531:
are partially due to diffraction, according to some analyses.
413:. The quantum approach has some striking similarities to the
5641:, Cham: Springer International Publishing, pp. 15–17,
682:
Simulated diffraction spikes in hexagonal telescope mirrors
432:
approximation of the Kirchhoff equation (applicable to the
6396:
The Feynman Lectures on Physics Vol. I Ch. 30: Diffraction
6229:(4th ed.). United States of America: Addison Wesley.
5530:
Kokkotas, Kostas D. (2003). "Gravitational Wave Physics".
4722:
of the imaging lens (e.g., of a telescope's main mirror).
1424:{\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}
559:
Pixels on smart phone screen acting as diffraction grating
5788:
Philosophical Transactions of the Royal Society of London
727:
to the resolution of a camera, telescope, or microscope.
2238:
The wave that emerges from a point source has amplitude
748:
2D Single-slit diffraction with width changing animation
6169:
Halliday, David; Resnick, Robert; Walker, Jerl (2005),
5711:
Memoires pour l'histoire des sciences et des beaux arts
5679:(Bologna ("Bonomia"), (Italy): Vittorio Bonati, 1665),
4613:) of the imaging optics; this is strictly accurate for
6250:
Ayahiko Ichimiya; Philip I. Cohen (13 December 2004).
1761:
Diffraction of a red laser using a diffraction grating
1753:
2-slit (top) and 5-slit diffraction of red laser light
1356:
1320:
951:{\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}
177:), a complex pattern of varying intensity can result.
6422:
5714:. Paris: Impr. de S. A. S.; Chez E. Ganeau. pp.
5502:. Springer Science & Business Media. p. 14.
5430:(Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665),
5211:
5172:
5152:
5132:
5112:
5068:
4994:
4970:
4933:
4704:
4653:
4619:
4599:
4579:
4555:
4535:
4497:
4185:
4127:
4076:
3838:
3710:
3637:
3420:
3290:
3157:
3079:
2848:
2795:
2660:
2634:
2607:
2539:
2503:
2402:
2363:
2297:
2264:
2244:
2205:
2174:
2154:
2134:
2017:
1945:
1925:
1898:
1817:
1724:
1606:
1575:
1542:
1514:
1491:
1462:
1390:
1292:
1263:
1234:
1141:
1110:
1061:
1034:
1011:
984:
964:
918:
891:
847:
816:
787:
123:
99:
5879:"Fresnel's prize memoir on the diffraction of light"
5496:
Suryanarayana, C.; Norton, M. Grant (29 June 2013).
4835:
extended the (singular) Keller coefficients via the
730:
Other examples of diffraction are considered below.
663:
are diffraction patterns caused due to non-circular
5988:"Food Explainer: Why Is Some Deli Meat Iridescent?"
5873:, vol. 1 (Paris: Imprimerie Impériale, 1866),
5840:, vol. 1 (Paris: Imprimerie Impériale, 1866),
2592:{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}
6170:
5217:
5178:
5158:
5138:
5118:
5098:
5000:
4976:
4956:
4710:
4690:
4631:
4605:
4585:
4561:
4541:
4521:
4397:
4168:
4113:
4060:
3824:
3696:
3621:
3403:
3276:
3141:
3055:
2831:
2779:
2642:
2620:
2591:
2525:
2475:
2380:
2349:
2272:
2250:
2218:
2191:
2160:
2140:
2120:
1951:
1931:
1911:
1884:
1737:
1707:
1588:
1554:
1526:
1497:
1474:
1423:
1375:
1342:
1304:
1276:
1249:
1220:
1116:
1096:
1047:
1017:
997:
970:
950:
904:
877:
830:
796:
129:
105:
93:Infinitely many points (three shown) along length
6048:Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964).
4266:
4265:
3919:
3918:
3813:
3789:
3750:
3717:
3688:
3665:
3581:
3501:
3500:
3390:
3130:
3107:
2898:
2897:
1959:is an integer which can be positive or negative.
885:so that the minimum intensity occurs at an angle
145:, the diffraction phenomenon is described by the
6177:(7th ed.), USA: John Wiley and Sons, Inc.,
5225:, allowing one to deduce the crystal structure.
4169:{\displaystyle k_{y}=k\sin \theta \sin \phi \,,}
638:, as seen from a plane on the underlying clouds.
5532:Encyclopedia of Physical Science and Technology
4814:Knife-edge diffraction is an outgrowth of the "
5756:, ed. Stephen Jordan Rigaud (Oxford, England:
3069:On the calculation of Fraunhofer region fields
1097:{\displaystyle d\,\sin \theta _{n}=n\lambda ,}
6195:
6193:
5952:The Mathematical Theory of Huygens' Principle
4799:is a truncation of a portion of the incident
4114:{\displaystyle k_{x}=k\sin \theta \cos \phi }
1919:is the angle at which the light is incident,
1484:the intensity will have little dependency on
325:and reinvigorated by Young, against Newton's
277:studied these effects and attributed them to
117:, producing a continuously varying intensity
8:
2628:and the field point is located at the point
2487:del in cylindrical and spherical coordinates
604:and internal reflection within the droplet.
6253:Reflection High-Energy Electron Diffraction
4643:case). In object space, the corresponding
1939:is the separation of grating elements, and
878:{\displaystyle {\frac {d\sin(\theta )}{2}}}
600:of the light that forms the glory involves
4957:{\displaystyle \lambda ={\frac {h}{p}}\,,}
1966:of diffraction and interference patterns.
764:Graph and image of single-slit diffraction
6369:
5808:
5760:, 1841), vol. 2, pp. 251–255, especially
5577:
5210:
5171:
5166:is the angle of the diffracted wave, and
5151:
5131:
5111:
5099:{\displaystyle m\lambda =2d\sin \theta ,}
5067:
4993:
4969:
4950:
4940:
4932:
4703:
4675:
4652:
4618:
4598:
4578:
4554:
4534:
4496:
4380:
4368:
4349:
4328:
4314:
4272:
4271:
4237:
4236:
4207:
4201:
4184:
4162:
4132:
4126:
4081:
4075:
4043:
4031:
3967:
3925:
3924:
3890:
3889:
3860:
3854:
3837:
3808:
3807:
3784:
3783:
3745:
3744:
3712:
3711:
3709:
3683:
3682:
3660:
3659:
3639:
3636:
3604:
3592:
3576:
3575:
3563:
3549:
3507:
3506:
3472:
3471:
3442:
3436:
3419:
3385:
3384:
3372:
3358:
3329:
3323:
3308:
3302:
3297:
3289:
3263:
3254:
3245:
3240:
3226:
3217:
3208:
3203:
3196:
3190:
3175:
3169:
3164:
3156:
3125:
3124:
3102:
3101:
3081:
3078:
3038:
3026:
3018:
3009:
3000:
2995:
2981:
2972:
2963:
2958:
2951:
2945:
2904:
2903:
2869:
2868:
2847:
2801:
2800:
2794:
2766:
2757:
2748:
2743:
2729:
2720:
2711:
2706:
2699:
2693:
2678:
2672:
2667:
2659:
2635:
2633:
2609:
2606:
2561:
2555:
2538:
2508:
2502:
2449:
2435:
2429:
2419:
2407:
2401:
2370:
2362:
2336:
2318:
2302:
2296:
2265:
2263:
2243:
2210:
2204:
2181:
2173:
2153:
2133:
2109:
2058:
2048:
2037:
2016:
1944:
1924:
1903:
1897:
1858:
1853:
1837:
1832:
1816:
1729:
1723:
1691:
1651:
1637:
1632:
1626:
1605:
1580:
1574:
1541:
1513:
1490:
1461:
1403:
1389:
1363:
1355:
1330:
1319:
1291:
1268:
1262:
1233:
1186:
1172:
1167:
1161:
1140:
1109:
1076:
1065:
1060:
1039:
1033:
1010:
989:
983:
963:
933:
922:
917:
896:
890:
848:
846:
820:
815:
786:
122:
98:
5146:is the distance between crystal planes,
4465:
2148:is the radius of the circular aperture,
1811:which are given by the grating equation
1284:is the intensity at the central maximum
220:also demonstrates that matter possesses
184:travels through a medium with a varying
153:as a collection of individual spherical
149:that treats each point in a propagating
6429:
5950:Baker, B.B. & Copson, E.T. (1939),
5499:X-Ray Diffraction: A Practical Approach
5419:
4841:
4522:{\displaystyle \Delta x=1.22\lambda N,}
4428:The way in which the beam profile of a
3411:as can be seen in the adjacent figure.
2832:{\displaystyle E_{\mathrm {inc} }(x,y)}
2497:(and using the physics time convention
1343:{\textstyle \theta =-{\frac {\pi }{2}}}
469:
6410:International Union of Crystallography
6144:
6142:
6092:IEEE Antennas and Propagation Magazine
6002:from the original on 10 September 2013
1376:{\textstyle \theta ={\frac {\pi }{2}}}
341:Single-slit diffraction in a circular
192:travels through a medium with varying
5865:(for 1821 & 1822, printed 1826),
4831:(GTD). In 1974, Prabhakar Pathak and
2381:{\displaystyle \delta (\mathbf {r} )}
2280:that is given by the solution of the
113:project phase contributions from the
7:
5986:Arumugam, Nadia (9 September 2013).
5871:Oeuvres complètes d'Augustin Fresnel
5838:Oeuvres complètes d'Augustin Fresnel
4851:Diffraction on a sharp metallic edge
2011:in intensity with angle is given by
998:{\displaystyle \theta _{\text{min}}}
905:{\displaystyle \theta _{\text{min}}}
356:propagate; this is described by the
6417:Using a cd as a diffraction grating
6024:Dynamic fields and waves of physics
5848:submitted on 15 October 1815.)
4549:is the wavelength of the light and
2654:(for arbitrary source location) as
2650:, then we may represent the scalar
1257:is the intensity at a given angle,
362:principle of superposition of waves
6270:from the original on 16 July 2017.
4498:
4279:
4276:
4273:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4186:
3932:
3929:
3926:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3839:
3514:
3511:
3508:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3421:
2911:
2908:
2905:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2849:
2808:
2805:
2802:
2442:
2432:
2404:
2299:
1738:{\displaystyle \theta _{\text{i}}}
1589:{\displaystyle \theta _{\text{i}}}
1440:This analysis applies only to the
440:approximation (applicable to the
352:diffraction arises because of how
25:
5635:"The Early Theory of Diffraction"
4829:geometrical theory of diffraction
2392:(a.k.a. scalar Laplacian) in the
1988:A computer-generated image of an
27:Phenomenon of the motion of waves
6492:
6480:
6468:
6456:
6444:
6432:
6346:"Diffraction before destruction"
6050:"Self-Trapping of Optical Beams"
5966:"Optical effects on spider webs"
5889:, vol. 11 (1819), pp.
5887:Annales de Chimie et de Physique
5834:Annales de Chimie et de Physique
5477:MacMillan Company, New York 1899
4856:
4844:
3810:
3786:
3747:
3714:
3685:
3662:
3640:
3578:
3564:
3387:
3373:
3309:
3298:
3255:
3246:
3218:
3209:
3176:
3165:
3127:
3104:
3082:
3010:
3001:
2973:
2964:
2789:Therefore, if an electric field
2758:
2749:
2721:
2712:
2679:
2668:
2636:
2610:
2371:
2337:
2266:
1555:{\displaystyle \theta \approx 0}
1383:and conservation of energy, and
1025:is the wavelength of the light.
627:
611:
582:on a credit card is an example.
564:
552:
520:
508:
496:
484:
472:
196:– all waves diffract, including
180:These effects also occur when a
5781:Thomas Young (1 January 1804).
5343:Dynamical theory of diffraction
4818:problem", originally solved by
1124:is an integer other than zero.
588:by small particles can cause a
66:into the region of geometrical
6256:. Cambridge University Press.
5540:10.1016/B0-12-227410-5/00300-8
5274:Diffraction before destruction
4363:
4321:
4307:
4285:
4195:
4189:
4026:
3986:
3960:
3938:
3848:
3842:
3587:
3559:
3542:
3520:
3430:
3424:
3396:
3368:
3317:
3303:
3294:
3264:
3241:
3227:
3204:
3184:
3170:
3161:
3019:
2996:
2982:
2959:
2939:
2917:
2858:
2852:
2826:
2814:
2767:
2744:
2730:
2707:
2687:
2673:
2664:
2549:
2543:
2526:{\displaystyle e^{-i\omega t}}
2467:
2458:
2375:
2367:
2341:
2333:
2192:{\displaystyle 2\pi /\lambda }
2082:
2064:
2027:
2021:
1697:
1666:
1616:
1610:
1244:
1238:
1151:
1145:
866:
860:
422:Kirchhoff diffraction equation
1:
6202:Introduction to Modern Optics
5398:Schaefer–Bergmann diffraction
4837:uniform theory of diffraction
4593:divided by aperture diameter
4413:far-field diffraction pattern
2621:{\displaystyle \mathbf {r} '}
1527:{\displaystyle d\gg \lambda }
1475:{\displaystyle d\ll \lambda }
586:Diffraction in the atmosphere
6402:"Scattering and diffraction"
5720:grimaldi diffraction 0–1800.
5328:Diffraction vs. interference
5313:Coherent diffraction imaging
4436:wave front, it approximates
2643:{\displaystyle \mathbf {r} }
2273:{\displaystyle \mathbf {r} }
1131:can be calculated using the
6152:(North-Holland, Amsterdam)
5732:Sir David Brewster (1831).
5647:10.1007/978-3-030-26699-8_2
5186:is an integer known as the
4456:Diffraction-limited imaging
4424:Propagation of a laser beam
2495:spherical coordinate system
2394:spherical coordinate system
1912:{\displaystyle \theta _{i}}
1048:{\displaystyle \theta _{n}}
327:corpuscular theory of light
258:, who also coined the term
6536:
6074:10.1103/PhysRevLett.13.479
6027:. CRC Press. p. 102.
5954:, Oxford, pp. 36–40.
5881:, in H. Crew (ed.),
5708:Jean Louis Aubert (1760).
5675:Francesco Maria Grimaldi,
5426:Francesco Maria Grimaldi,
5239:
5027:
4908:
4772:
4749:
4462:Diffraction-limited system
4459:
1977:
1777:
1435:unnormalized sinc function
1250:{\displaystyle I(\theta )}
978:is the width of the slit,
831:{\displaystyle \lambda /2}
737:
29:
4805:Huygens–Fresnel principle
4768:
1305:{\displaystyle \theta =0}
771:Huygens–Fresnel principle
446:path integral formulation
415:Huygens-Fresnel principle
386:modern quantum mechanical
358:Huygens–Fresnel principle
147:Huygens–Fresnel principle
6200:Grant R. Fowles (1975).
6104:10.1109/MAP.2013.6586622
5190:of the diffracted beam.
5119:{\displaystyle \lambda }
4542:{\displaystyle \lambda }
2287:for a point source (the
1569:When the incident angle
1018:{\displaystyle \lambda }
398:probability distribution
256:Francesco Maria Grimaldi
137:on the registering plate
76:Francesco Maria Grimaldi
74:wave. Italian scientist
30:Not to be confused with
6204:. Courier Corporation.
6120:Proceedings of the IEEE
6054:Physical Review Letters
5877:; partly translated as
5758:Oxford University Press
5298:Atmospheric diffraction
5159:{\displaystyle \theta }
4905:Matter wave diffraction
4718:is the diameter of the
1498:{\displaystyle \theta }
734:Single-slit diffraction
171:closely spaced openings
130:{\displaystyle \theta }
6362:10.1098/rstb.2013.0313
6225:Hecht, Eugene (2002).
6173:Fundamental of Physics
6148:John M. Cowley (1975)
6133:10.1109/PROC.1974.9651
6021:Andrew Norton (2000).
5810:10.1098/rstl.1804.0001
5457:
5449:
5353:Fraunhofer diffraction
5333:Diffractive solar sail
5318:Diffraction from slits
5219:
5180:
5160:
5140:
5120:
5100:
5047:
5002:
4978:
4958:
4878:
4797:knife-edge diffraction
4712:
4692:
4633:
4632:{\displaystyle N\gg 1}
4607:
4587:
4563:
4543:
4523:
4484:
4399:
4170:
4115:
4062:
3826:
3698:
3623:
3405:
3278:
3143:
3070:
3057:
2833:
2781:
2644:
2622:
2593:
2527:
2477:
2382:
2351:
2274:
2252:
2220:
2193:
2162:
2142:
2122:
2000:
1992:
1953:
1933:
1913:
1886:
1798:
1770:
1762:
1754:
1739:
1709:
1590:
1556:
1528:
1499:
1476:
1446:Fraunhofer diffraction
1425:
1377:
1344:
1306:
1278:
1251:
1222:
1133:Fraunhofer diffraction
1118:
1098:
1049:
1019:
999:
972:
952:
906:
879:
832:
798:
765:
757:
749:
717:
683:
675:
657:
430:Fraunhofer diffraction
403:double-slit experiment
345:
248:
138:
131:
107:
56:
5596:10.1038/nnano.2012.34
5566:Nature Nanotechnology
5450:
5442:
5373:Point spread function
5293:Angle-sensitive pixel
5220:
5197:or matter waves like
5181:
5161:
5141:
5121:
5101:
5037:
5028:Further information:
5003:
4979:
4959:
4926:de Broglie wavelength
4876:
4736:microscope objectives
4713:
4693:
4634:
4608:
4588:
4564:
4544:
4524:
4469:
4400:
4171:
4116:
4063:
3827:
3699:
3624:
3406:
3279:
3144:
3068:
3058:
2834:
2782:
2645:
2623:
2594:
2528:
2478:
2383:
2352:
2275:
2253:
2251:{\displaystyle \psi }
2221:
2219:{\displaystyle J_{1}}
2194:
2163:
2143:
2123:
1998:
1987:
1954:
1934:
1914:
1887:
1796:
1768:
1760:
1752:
1740:
1710:
1591:
1557:
1529:
1500:
1477:
1426:
1378:
1345:
1307:
1279:
1277:{\displaystyle I_{0}}
1252:
1223:
1119:
1099:
1050:
1020:
1000:
973:
953:
907:
880:
833:
799:
797:{\displaystyle 2\pi }
763:
755:
747:
740:Diffraction formalism
715:
709:and other obstacles.
681:
673:
651:
340:
311:Augustin-Jean Fresnel
303:celebrated experiment
243:
206:electromagnetic waves
132:
108:
92:
42:
5909:Traité de la lumiere
5906:Christiaan Huygens,
5735:A Treatise on Optics
5348:Electron diffraction
5280:free-electron lasers
5230:electron diffraction
5209:
5170:
5150:
5130:
5110:
5066:
4992:
4968:
4931:
4919:Electron diffraction
4702:
4651:
4617:
4597:
4577:
4553:
4533:
4495:
4183:
4125:
4074:
3836:
3708:
3635:
3418:
3288:
3155:
3077:
2846:
2793:
2658:
2632:
2605:
2537:
2501:
2400:
2361:
2295:
2262:
2242:
2203:
2172:
2152:
2132:
2015:
1943:
1923:
1896:
1815:
1722:
1604:
1573:
1540:
1512:
1489:
1460:
1388:
1354:
1318:
1290:
1261:
1232:
1139:
1108:
1059:
1032:
1009:
982:
962:
916:
889:
845:
814:
785:
396:that determines the
392:is described by its
222:wave-like properties
121:
97:
6299:2000Natur.406..752N
6150:Diffraction physics
6088:Rahmat-Samii, Yahya
6066:1964PhRvL..13..479C
5964:Dietrich Zawischa.
5846:"First Memoir"
5844:. (Revision of the
5801:1804RSPT...94....1Y
5588:2012NatNa...7..297J
5403:Thinned-array curse
5242:Coherence (physics)
5126:is the wavelength,
5056:diffraction grating
4915:Neutron diffraction
4780:Babinet's principle
4775:Babinet's principle
4769:Babinet's principle
1797:Diffraction grating
1780:Diffraction grating
1774:Diffraction grating
543:diffraction grating
438:Fresnel diffraction
405:first performed by
295:diffraction grating
198:gravitational waves
175:diffraction grating
45:diffraction pattern
6520:Physical phenomena
6356:(1647): 20130313.
5927:2016-12-01 at the
5916:2016-06-16 at the
5767:2016-12-01 at the
5686:2016-12-01 at the
5473:2016-12-01 at the
5437:2016-12-01 at the
5378:Powder diffraction
5215:
5176:
5156:
5136:
5116:
5096:
5048:
4998:
4974:
4954:
4879:
4833:Robert Kouyoumjian
4760:seen when using a
4740:numerical aperture
4728:Rayleigh criterion
4708:
4688:
4645:angular resolution
4629:
4603:
4583:
4559:
4539:
4519:
4485:
4395:
4264:
4166:
4111:
4058:
3917:
3822:
3694:
3619:
3499:
3401:
3274:
3139:
3071:
3053:
2896:
2829:
2777:
2640:
2618:
2589:
2523:
2473:
2378:
2347:
2289:Helmholtz equation
2270:
2248:
2216:
2189:
2158:
2138:
2118:
2001:
1993:
1949:
1929:
1909:
1882:
1799:
1771:
1763:
1755:
1735:
1705:
1586:
1564:geometrical optics
1552:
1524:
1495:
1472:
1421:
1373:
1340:
1302:
1274:
1247:
1218:
1114:
1094:
1045:
1015:
995:
968:
948:
902:
875:
828:
794:
766:
758:
750:
718:
684:
676:
661:Diffraction spikes
658:
444:) and the Feynman
424:(derived from the
346:
323:Christiaan Huygens
297:to be discovered.
249:
194:acoustic impedance
139:
127:
103:
57:
6293:(6797): 752–757.
6263:978-0-521-45373-8
6236:978-0-8053-8566-3
6211:978-0-486-65957-2
6184:978-0-471-23231-5
6127:(11): 1448–1461.
6034:978-0-7503-0719-2
5656:978-3-030-26699-8
5509:978-1-4899-0148-4
5408:X-ray diffraction
5323:Diffraction spike
5308:Cloud iridescence
5256:atomic transition
5218:{\displaystyle d}
5179:{\displaystyle m}
5139:{\displaystyle d}
5052:Bragg diffraction
5030:Bragg diffraction
5024:Bragg diffraction
5001:{\displaystyle p}
4977:{\displaystyle h}
4948:
4820:Arnold Sommerfeld
4793:knife-edge effect
4711:{\displaystyle D}
4683:
4606:{\displaystyle D}
4586:{\displaystyle f}
4562:{\displaystyle N}
4409:Fourier transform
4232:
4230:
4178:Fourier transform
3885:
3883:
3816:
3792:
3782:
3770:
3753:
3720:
3691:
3668:
3584:
3467:
3465:
3393:
3352:
3269:
3133:
3110:
3024:
2944:
2864:
2772:
2584:
2456:
2427:
2161:{\displaystyle k}
2141:{\displaystyle a}
2103:
1974:Circular aperture
1952:{\displaystyle m}
1932:{\displaystyle d}
1794:
1732:
1694:
1664:
1583:
1453:intensity profile
1419:
1371:
1338:
1199:
1129:intensity profile
1117:{\displaystyle n}
992:
971:{\displaystyle d}
936:
899:
873:
725:fundamental limit
652:The bright spot (
527:Colors seen in a
350:classical physics
218:quantum mechanics
143:classical physics
106:{\displaystyle d}
16:(Redirected from
6527:
6497:
6496:
6495:
6485:
6484:
6483:
6473:
6472:
6461:
6460:
6459:
6449:
6448:
6437:
6436:
6428:
6413:
6384:
6383:
6373:
6341:
6335:
6334:
6307:10.1038/35021099
6278:
6272:
6271:
6247:
6241:
6240:
6222:
6216:
6215:
6197:
6188:
6187:
6176:
6166:
6160:
6146:
6137:
6136:
6114:
6108:
6107:
6084:
6078:
6077:
6045:
6039:
6038:
6018:
6012:
6011:
6009:
6007:
5983:
5977:
5976:
5974:
5972:
5961:
5955:
5948:
5942:
5904:
5898:
5875:pp. 247–364
5867:pp. 339–475
5864:
5855:
5849:
5830:
5824:
5822:
5812:
5778:
5772:
5750:
5744:
5743:
5729:
5723:
5722:
5705:
5699:
5673:
5667:
5666:
5665:
5663:
5630:
5624:
5623:
5581:
5560:
5554:
5553:
5527:
5521:
5520:
5518:
5516:
5493:
5487:
5484:
5478:
5466:Cajori, Florian
5464:
5458:
5424:
5252:coherence length
5224:
5222:
5221:
5216:
5185:
5183:
5182:
5177:
5165:
5163:
5162:
5157:
5145:
5143:
5142:
5137:
5125:
5123:
5122:
5117:
5105:
5103:
5102:
5097:
5007:
5005:
5004:
4999:
4983:
4981:
4980:
4975:
4963:
4961:
4960:
4955:
4949:
4941:
4860:
4848:
4825:Joseph B. Keller
4746:Speckle patterns
4738:require a large
4717:
4715:
4714:
4709:
4697:
4695:
4694:
4689:
4684:
4676:
4638:
4636:
4635:
4630:
4612:
4610:
4609:
4604:
4592:
4590:
4589:
4584:
4568:
4566:
4565:
4560:
4548:
4546:
4545:
4540:
4528:
4526:
4525:
4520:
4404:
4402:
4401:
4396:
4391:
4379:
4367:
4366:
4362:
4354:
4353:
4341:
4333:
4332:
4306:
4295:
4284:
4283:
4282:
4263:
4262:
4231:
4229:
4218:
4217:
4202:
4175:
4173:
4172:
4167:
4137:
4136:
4120:
4118:
4117:
4112:
4086:
4085:
4067:
4065:
4064:
4059:
4054:
4042:
4030:
4029:
4025:
4005:
3959:
3948:
3937:
3936:
3935:
3916:
3915:
3884:
3882:
3871:
3870:
3855:
3831:
3829:
3828:
3823:
3818:
3817:
3809:
3794:
3793:
3785:
3780:
3768:
3755:
3754:
3746:
3722:
3721:
3713:
3703:
3701:
3700:
3695:
3693:
3692:
3684:
3681:
3670:
3669:
3661:
3658:
3647:
3643:
3628:
3626:
3625:
3620:
3615:
3603:
3591:
3590:
3586:
3585:
3577:
3571:
3567:
3541:
3530:
3519:
3518:
3517:
3498:
3497:
3466:
3464:
3453:
3452:
3437:
3410:
3408:
3407:
3402:
3400:
3399:
3395:
3394:
3386:
3380:
3376:
3353:
3351:
3340:
3339:
3324:
3316:
3312:
3306:
3301:
3283:
3281:
3280:
3275:
3270:
3268:
3267:
3262:
3258:
3249:
3244:
3232:
3231:
3230:
3225:
3221:
3212:
3207:
3191:
3183:
3179:
3173:
3168:
3148:
3146:
3145:
3140:
3135:
3134:
3126:
3123:
3112:
3111:
3103:
3100:
3089:
3085:
3062:
3060:
3059:
3054:
3049:
3037:
3025:
3023:
3022:
3017:
3013:
3004:
2999:
2987:
2986:
2985:
2980:
2976:
2967:
2962:
2946:
2942:
2938:
2927:
2916:
2915:
2914:
2895:
2894:
2841:surface integral
2838:
2836:
2835:
2830:
2813:
2812:
2811:
2786:
2784:
2783:
2778:
2773:
2771:
2770:
2765:
2761:
2752:
2747:
2735:
2734:
2733:
2728:
2724:
2715:
2710:
2694:
2686:
2682:
2676:
2671:
2652:Green's function
2649:
2647:
2646:
2641:
2639:
2627:
2625:
2624:
2619:
2617:
2613:
2598:
2596:
2595:
2590:
2585:
2583:
2572:
2571:
2556:
2532:
2530:
2529:
2524:
2522:
2521:
2491:Green's function
2482:
2480:
2479:
2474:
2457:
2455:
2454:
2453:
2440:
2439:
2430:
2428:
2420:
2412:
2411:
2390:Laplace operator
2387:
2385:
2384:
2379:
2374:
2356:
2354:
2353:
2348:
2340:
2323:
2322:
2307:
2306:
2282:frequency domain
2279:
2277:
2276:
2271:
2269:
2257:
2255:
2254:
2249:
2234:General aperture
2225:
2223:
2222:
2217:
2215:
2214:
2198:
2196:
2195:
2190:
2185:
2167:
2165:
2164:
2159:
2147:
2145:
2144:
2139:
2127:
2125:
2124:
2119:
2114:
2113:
2108:
2104:
2102:
2085:
2063:
2062:
2049:
2042:
2041:
1958:
1956:
1955:
1950:
1938:
1936:
1935:
1930:
1918:
1916:
1915:
1910:
1908:
1907:
1891:
1889:
1888:
1883:
1869:
1865:
1864:
1863:
1862:
1843:
1842:
1841:
1795:
1746:
1744:
1742:
1741:
1736:
1734:
1733:
1730:
1714:
1712:
1711:
1706:
1704:
1700:
1696:
1695:
1692:
1665:
1660:
1652:
1642:
1641:
1631:
1630:
1595:
1593:
1592:
1587:
1585:
1584:
1581:
1561:
1559:
1558:
1553:
1535:
1533:
1531:
1530:
1525:
1506:
1504:
1502:
1501:
1496:
1483:
1481:
1479:
1478:
1473:
1432:
1430:
1428:
1427:
1422:
1420:
1415:
1404:
1382:
1380:
1379:
1374:
1372:
1364:
1349:
1347:
1346:
1341:
1339:
1331:
1313:
1311:
1309:
1308:
1303:
1283:
1281:
1280:
1275:
1273:
1272:
1256:
1254:
1253:
1248:
1227:
1225:
1224:
1219:
1214:
1210:
1200:
1195:
1187:
1177:
1176:
1166:
1165:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1081:
1080:
1054:
1052:
1051:
1046:
1044:
1043:
1024:
1022:
1021:
1016:
1004:
1002:
1001:
996:
994:
993:
990:
977:
975:
974:
969:
957:
955:
954:
949:
938:
937:
934:
911:
909:
908:
903:
901:
900:
897:
884:
882:
881:
876:
874:
869:
849:
839:
837:
835:
834:
829:
824:
803:
801:
800:
795:
631:
615:
568:
556:
524:
512:
500:
488:
476:
454:boundary element
368:for a secondary
186:refractive index
136:
134:
133:
128:
112:
110:
109:
104:
78:coined the word
55:in another plate
21:
6535:
6534:
6530:
6529:
6528:
6526:
6525:
6524:
6505:
6504:
6503:
6493:
6491:
6481:
6479:
6467:
6457:
6455:
6443:
6431:
6423:
6406:Crystallography
6400:
6392:
6387:
6343:
6342:
6338:
6280:
6279:
6275:
6264:
6249:
6248:
6244:
6237:
6224:
6223:
6219:
6212:
6199:
6198:
6191:
6185:
6168:
6167:
6163:
6147:
6140:
6116:
6115:
6111:
6086:
6085:
6081:
6060:(15): 479–482.
6047:
6046:
6042:
6035:
6020:
6019:
6015:
6005:
6003:
5996:The Slate Group
5985:
5984:
5980:
5970:
5968:
5963:
5962:
5958:
5949:
5945:
5929:Wayback Machine
5918:Wayback Machine
5905:
5901:
5869:; reprinted in
5862:
5856:
5852:
5842:pp. 89–122
5831:
5827:
5780:
5779:
5775:
5769:Wayback Machine
5751:
5747:
5731:
5730:
5726:
5707:
5706:
5702:
5688:Wayback Machine
5674:
5670:
5661:
5659:
5657:
5632:
5631:
5627:
5562:
5561:
5557:
5550:
5529:
5528:
5524:
5514:
5512:
5510:
5495:
5494:
5490:
5485:
5481:
5475:Wayback Machine
5465:
5461:
5439:Wayback Machine
5425:
5421:
5417:
5412:
5303:Brocken spectre
5288:
5276:
5271:
5244:
5238:
5207:
5206:
5168:
5167:
5148:
5147:
5128:
5127:
5108:
5107:
5064:
5063:
5042:, each dot (or
5032:
5026:
5015:Diffraction of
4990:
4989:
4986:Planck constant
4966:
4965:
4929:
4928:
4921:
4907:
4890:of the angles.)
4871:
4864:
4861:
4852:
4849:
4789:
4777:
4771:
4758:speckle pattern
4754:
4752:Speckle pattern
4748:
4700:
4699:
4649:
4648:
4615:
4614:
4595:
4594:
4575:
4574:
4551:
4550:
4531:
4530:
4493:
4492:
4464:
4458:
4426:
4384:
4372:
4355:
4345:
4334:
4324:
4310:
4299:
4288:
4267:
4219:
4203:
4181:
4180:
4128:
4123:
4122:
4077:
4072:
4071:
4047:
4035:
4018:
3998:
3963:
3952:
3941:
3920:
3872:
3856:
3834:
3833:
3706:
3705:
3674:
3651:
3638:
3633:
3632:
3608:
3596:
3562:
3545:
3534:
3523:
3502:
3454:
3438:
3416:
3415:
3371:
3354:
3341:
3325:
3307:
3286:
3285:
3253:
3233:
3216:
3192:
3174:
3153:
3152:
3116:
3093:
3080:
3075:
3074:
3042:
3030:
3008:
2988:
2971:
2947:
2931:
2920:
2899:
2844:
2843:
2796:
2791:
2790:
2756:
2736:
2719:
2695:
2677:
2656:
2655:
2630:
2629:
2608:
2603:
2602:
2573:
2557:
2535:
2534:
2504:
2499:
2498:
2493:, which in the
2445:
2441:
2431:
2403:
2398:
2397:
2359:
2358:
2314:
2298:
2293:
2292:
2260:
2259:
2240:
2239:
2236:
2228:Bessel function
2206:
2201:
2200:
2170:
2169:
2150:
2149:
2130:
2129:
2086:
2054:
2050:
2044:
2043:
2033:
2013:
2012:
1982:
1976:
1941:
1940:
1921:
1920:
1899:
1894:
1893:
1854:
1833:
1825:
1821:
1813:
1812:
1810:
1784:
1782:
1776:
1725:
1720:
1719:
1718:
1687:
1653:
1650:
1646:
1633:
1622:
1602:
1601:
1576:
1571:
1570:
1538:
1537:
1510:
1509:
1508:
1487:
1486:
1485:
1458:
1457:
1456:
1405:
1386:
1385:
1384:
1352:
1351:
1316:
1315:
1288:
1287:
1285:
1264:
1259:
1258:
1230:
1229:
1188:
1185:
1181:
1168:
1157:
1137:
1136:
1106:
1105:
1072:
1057:
1056:
1035:
1030:
1029:
1007:
1006:
985:
980:
979:
960:
959:
929:
914:
913:
892:
887:
886:
850:
843:
842:
812:
811:
810:
783:
782:
742:
736:
688:speckle pattern
643:
642:
641:
640:
639:
632:
624:
623:
616:
576:
575:
574:
573:
572:
569:
561:
560:
557:
539:
532:
525:
516:
513:
504:
501:
492:
489:
480:
477:
376:as well as the
335:
281:of light rays.
238:
216:. Furthermore,
119:
118:
95:
94:
35:
28:
23:
22:
15:
12:
11:
5:
6533:
6531:
6523:
6522:
6517:
6507:
6506:
6502:
6501:
6489:
6477:
6465:
6453:
6441:
6421:
6420:
6414:
6398:
6391:
6390:External links
6388:
6386:
6385:
6336:
6273:
6262:
6242:
6235:
6217:
6210:
6189:
6183:
6161:
6138:
6109:
6079:
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5978:
5956:
5943:
5899:
5850:
5825:
5773:
5745:
5724:
5700:
5668:
5655:
5625:
5572:(5): 297–300.
5555:
5548:
5522:
5508:
5488:
5479:
5459:
5418:
5416:
5413:
5411:
5410:
5405:
5400:
5395:
5390:
5385:
5380:
5375:
5370:
5365:
5363:Fresnel number
5360:
5358:Fresnel imager
5355:
5350:
5345:
5340:
5338:Diffractometer
5335:
5330:
5325:
5320:
5315:
5310:
5305:
5300:
5295:
5289:
5287:
5284:
5275:
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5240:Main article:
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4891:
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4865:
4862:
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4853:
4850:
4843:
4788:
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4773:Main article:
4770:
4767:
4750:Main article:
4747:
4744:
4720:entrance pupil
4707:
4687:
4682:
4679:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4628:
4625:
4622:
4602:
4582:
4573:(focal length
4558:
4538:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4460:Main article:
4457:
4454:
4425:
4422:
4418:Fourier optics
4394:
4390:
4387:
4383:
4378:
4375:
4371:
4365:
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4352:
4348:
4344:
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4331:
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4194:
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4188:
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4152:
4149:
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4080:
4057:
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4017:
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4008:
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4001:
3997:
3994:
3991:
3988:
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3875:
3869:
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3821:
3815:
3812:
3806:
3803:
3800:
3797:
3791:
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3779:
3776:
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3737:
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3728:
3725:
3719:
3716:
3690:
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3680:
3677:
3673:
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3646:
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3478:
3475:
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3357:
3350:
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3338:
3335:
3332:
3328:
3322:
3319:
3315:
3311:
3305:
3300:
3296:
3293:
3284:simplifies to
3273:
3266:
3261:
3257:
3252:
3248:
3243:
3239:
3236:
3229:
3224:
3220:
3215:
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3206:
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3178:
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3122:
3119:
3115:
3109:
3106:
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3092:
3088:
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3033:
3029:
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3016:
3012:
3007:
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2994:
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2984:
2979:
2975:
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2957:
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2941:
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2878:
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2804:
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2755:
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2746:
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2739:
2732:
2727:
2723:
2718:
2714:
2709:
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2702:
2698:
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2675:
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2663:
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2588:
2582:
2579:
2576:
2570:
2567:
2564:
2560:
2554:
2551:
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2545:
2542:
2520:
2517:
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2511:
2507:
2472:
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2460:
2452:
2448:
2444:
2438:
2434:
2426:
2423:
2418:
2415:
2410:
2406:
2396:simplifies to
2377:
2373:
2369:
2366:
2346:
2343:
2339:
2335:
2332:
2329:
2326:
2321:
2317:
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2310:
2305:
2301:
2268:
2247:
2235:
2232:
2213:
2209:
2188:
2184:
2180:
2177:
2157:
2137:
2117:
2112:
2107:
2101:
2098:
2095:
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2089:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2061:
2057:
2053:
2047:
2040:
2036:
2032:
2029:
2026:
2023:
2020:
1978:Main article:
1975:
1972:
1948:
1928:
1906:
1902:
1881:
1878:
1875:
1872:
1868:
1861:
1857:
1852:
1849:
1846:
1840:
1836:
1831:
1828:
1824:
1820:
1806:
1778:Main article:
1775:
1772:
1728:
1703:
1699:
1690:
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1680:
1677:
1674:
1671:
1668:
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1629:
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1621:
1618:
1615:
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1609:
1579:
1551:
1548:
1545:
1523:
1520:
1517:
1494:
1471:
1468:
1465:
1418:
1414:
1411:
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1402:
1399:
1396:
1393:
1370:
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1362:
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895:
872:
868:
865:
862:
859:
856:
853:
827:
823:
819:
793:
790:
738:Main article:
735:
732:
694:appears to be
633:
626:
625:
617:
610:
609:
608:
607:
606:
598:backscattering
570:
563:
562:
558:
551:
550:
549:
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533:
526:
519:
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478:
471:
450:finite element
370:spherical wave
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24:
14:
13:
10:
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6407:
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6159:
6158:0-444-10791-6
6155:
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5549:9780122274107
5545:
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5273:
5268:
5266:
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5173:
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5018:
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4834:
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4826:
4821:
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4806:
4802:
4798:
4794:
4787:"Knife edge"
4786:
4784:
4781:
4776:
4766:
4763:
4762:laser pointer
4759:
4753:
4745:
4743:
4741:
4737:
4732:
4729:
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4721:
4705:
4685:
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4672:
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4490:
4482:
4479:
4475:
4474:
4468:
4463:
4455:
4453:
4449:
4447:
4446:self-focusing
4443:
4439:
4438:Gaussian beam
4435:
4431:
4423:
4421:
4419:
4414:
4410:
4405:
4392:
4388:
4385:
4381:
4376:
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4342:
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3326:
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2725:
2716:
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2690:
2683:
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2653:
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2568:
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2492:
2488:
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2470:
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2461:
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2424:
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2416:
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2364:
2344:
2330:
2327:
2324:
2319:
2315:
2311:
2308:
2303:
2290:
2286:
2285:wave equation
2283:
2245:
2233:
2231:
2229:
2211:
2207:
2186:
2182:
2178:
2175:
2155:
2135:
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2110:
2105:
2099:
2096:
2093:
2090:
2087:
2079:
2076:
2073:
2070:
2067:
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2055:
2051:
2045:
2038:
2034:
2030:
2024:
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2010:
2006:
1997:
1991:
1986:
1981:
1973:
1971:
1967:
1965:
1960:
1946:
1926:
1904:
1900:
1879:
1876:
1873:
1870:
1866:
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1838:
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1829:
1826:
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1818:
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1781:
1773:
1767:
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1747:
1726:
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1688:
1684:
1681:
1678:
1675:
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1669:
1661:
1657:
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1647:
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1638:
1634:
1627:
1623:
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1613:
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1577:
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1518:
1515:
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1469:
1466:
1463:
1454:
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1447:
1443:
1438:
1436:
1433:which is the
1416:
1412:
1409:
1406:
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1394:
1391:
1368:
1365:
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1134:
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470:
468:
466:
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426:wave equation
423:
418:
416:
412:
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404:
399:
395:
391:
387:
382:
379:
375:
371:
367:
363:
359:
355:
351:
344:
339:
332:
330:
328:
324:
320:
316:
312:
308:
304:
300:
296:
292:
288:
284:
283:James Gregory
280:
276:
272:
268:
265:
261:
257:
253:
247:
242:
235:
233:
231:
230:semi-circular
225:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
178:
176:
172:
168:
164:
160:
156:
152:
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144:
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6276:
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6220:
6201:
6172:
6164:
6149:
6124:
6118:
6112:
6098:(3): 29–40.
6095:
6091:
6082:
6057:
6053:
6043:
6023:
6016:
6004:. Retrieved
5991:
5981:
5971:21 September
5969:. Retrieved
5959:
5951:
5946:
5936:
5932:
5908:
5902:
5886:
5882:
5870:
5861:, vol.
5858:
5853:
5837:
5833:
5828:
5792:
5786:
5776:
5753:
5748:
5734:
5727:
5719:
5710:
5703:
5695:
5691:
5676:
5671:
5660:, retrieved
5638:
5628:
5569:
5565:
5558:
5531:
5525:
5513:. Retrieved
5498:
5491:
5482:
5462:
5452:
5451:
5444:
5443:
5427:
5422:
5368:Fresnel zone
5277:
5269:Applications
5264:
5260:
5249:
5245:
5227:
5192:
5187:
5059:
5049:
5043:
5017:matter waves
5014:
4922:
4880:
4813:
4796:
4792:
4790:
4778:
4755:
4733:
4724:
4486:
4471:
4450:
4427:
4406:
4069:
3630:
3413:
3150:
3072:
2788:
2600:
2484:
2258:at location
2237:
2168:is equal to
2002:
1989:
1968:
1961:
1807:
1803:
1800:
1716:
1568:
1450:
1439:
1135:equation as
1126:
1027:
807:
775:
767:
729:
722:
719:
704:
685:
659:
644:
584:
577:
540:
462:
458:
419:
407:G. I. Taylor
394:wavefunction
383:
366:point source
347:
301:performed a
299:Thomas Young
278:
275:Isaac Newton
266:
259:
250:
226:
204:, and other
188:, or when a
179:
167:interference
140:
79:
59:
58:
36:
6515:Diffraction
6499:Outer space
6487:Spaceflight
6451:Mathematics
6006:9 September
5453:Translation
5383:Quasioptics
5060:Bragg's law
5040:Bragg's law
4911:Matter wave
4898:double-slit
4489:diffraction
4481:zeta Boötis
4478:binary star
4473:lucky image
4442:convex lens
3631:Now, since
1964:convolution
1598:path length
465:water waves
343:ripple tank
267:diffringere
262:, from the
260:diffraction
214:radio waves
202:water waves
80:diffraction
72:propagating
60:Diffraction
6509:Categories
6419:at YouTube
5415:References
5393:Reflection
5388:Refraction
5044:reflection
5038:Following
4909:See also:
4816:half-plane
4430:laser beam
1455:above, if
696:iridescent
654:Arago spot
602:refraction
529:spider web
442:near field
378:amplitudes
190:sound wave
182:light wave
163:wavelength
32:refraction
18:Diffracted
6463:Astronomy
6315:1476-4687
5819:110408369
5604:1748-3395
5579:1402.1867
5534:: 67–85.
5515:7 January
5236:Coherence
5203:electrons
5154:θ
5114:λ
5091:θ
5088:
5073:λ
4935:λ
4809:wavefront
4801:radiation
4678:λ
4667:θ
4664:
4658:≈
4655:θ
4624:≫
4537:λ
4511:λ
4499:Δ
4316:−
4234:∬
4224:π
4199:∝
4187:Ψ
4160:ϕ
4157:
4151:θ
4148:
4109:ϕ
4106:
4100:θ
4097:
4016:ϕ
4013:
3996:ϕ
3993:
3984:θ
3981:
3969:−
3887:∬
3877:π
3852:∝
3840:Ψ
3814:^
3805:θ
3802:
3790:^
3778:ϕ
3775:
3766:θ
3763:
3751:^
3742:ϕ
3739:
3733:θ
3730:
3718:^
3689:^
3666:^
3582:^
3573:⋅
3551:−
3469:∬
3459:π
3434:∝
3422:Ψ
3391:^
3382:⋅
3360:−
3346:π
3292:ψ
3251:−
3238:π
3214:−
3159:ψ
3131:^
3108:^
3006:−
2993:π
2969:−
2866:∬
2862:∝
2850:Ψ
2754:−
2741:π
2717:−
2662:ψ
2578:π
2541:ψ
2516:ω
2510:−
2465:ψ
2443:∂
2433:∂
2414:ψ
2405:∇
2365:δ
2331:δ
2325:ψ
2309:ψ
2300:∇
2246:ψ
2187:λ
2179:π
2100:θ
2097:
2080:θ
2077:
2025:θ
2009:variation
2005:Airy disk
1990:Airy disk
1980:Airy disk
1901:θ
1877:λ
1856:θ
1851:
1845:±
1835:θ
1830:
1727:θ
1689:θ
1685:
1679:±
1676:θ
1673:
1662:λ
1658:π
1644:
1614:θ
1578:θ
1547:≈
1544:θ
1522:λ
1519:≫
1493:θ
1470:λ
1467:≪
1451:From the
1442:far field
1410:
1395:
1366:π
1358:θ
1333:π
1328:−
1322:θ
1294:θ
1242:θ
1208:θ
1205:
1197:λ
1193:π
1179:
1149:θ
1089:λ
1074:θ
1070:
1055:given by
1037:θ
1013:λ
987:θ
943:λ
931:θ
927:
912:given by
894:θ
864:θ
858:
818:λ
792:π
692:deli meat
456:methods.
434:far field
333:Mechanism
279:inflexion
173:(e.g., a
151:wavefront
125:θ
115:wavefront
47:of a red
6380:24914146
6323:10963603
6268:Archived
6000:Archived
5925:Archived
5914:Archived
5795:: 1–16.
5765:Archived
5684:Archived
5681:pp. 1–11
5662:25 April
5612:22447163
5471:Archived
5445:Original
5435:Archived
5286:See also
5199:neutrons
5010:momentum
4869:Patterns
4641:paraxial
4571:f-number
4448:effect.
4434:coherent
4389:′
4377:′
4360:′
4339:′
4304:′
4293:′
4070:Letting
4052:′
4040:′
4023:′
4003:′
3957:′
3946:′
3679:′
3656:′
3645:′
3613:′
3601:′
3569:′
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3528:′
3378:′
3314:′
3260:′
3223:′
3181:′
3121:′
3098:′
3087:′
3047:′
3035:′
3015:′
2978:′
2936:′
2925:′
2763:′
2726:′
2684:′
2615:′
779:coherent
665:aperture
634:A solar
580:hologram
537:Examples
360:and the
252:Da Vinci
208:such as
159:coherent
155:wavelets
64:aperture
53:aperture
6439:Physics
6425:Portals
6371:4052855
6331:4300920
6295:Bibcode
6062:Bibcode
5797:Bibcode
5620:5918772
5584:Bibcode
5008:is the
4984:is the
4839:(UTD).
4569:is the
4476:of the
2007:. The
707:jetties
436:), the
428:), the
384:In the
236:History
6378:
6368:
6329:
6321:
6313:
6287:Nature
6260:
6233:
6227:Optics
6208:
6181:
6156:
6031:
5938:Traité
5895:337–78
5891:246–96
5817:
5762:p. 254
5653:
5618:
5610:
5602:
5546:
5506:
5432:page 2
5195:X-rays
5106:where
4964:where
4917:, and
4698:where
4529:where
3781:
3769:
2943:
2357:where
2128:where
1892:where
1228:where
1104:where
958:where
620:corona
618:Lunar
590:corona
390:photon
374:phases
210:X-rays
68:shadow
6475:Stars
6327:S2CID
5992:Slate
5922:p. 15
5815:S2CID
5616:S2CID
5574:arXiv
5201:(and
5188:order
4888:sines
2533:) is
2485:(See
2226:is a
1536:only
636:glory
594:glory
460:out.
354:waves
264:Latin
49:laser
6376:PMID
6319:PMID
6311:ISSN
6258:ISBN
6231:ISBN
6206:ISBN
6179:ISBN
6154:ISBN
6029:ISBN
6008:2013
5973:2007
5664:2024
5651:ISBN
5608:PMID
5600:ISSN
5544:ISBN
5517:2023
5504:ISBN
4988:and
4791:The
4756:The
4673:1.22
4508:1.22
4121:and
3704:and
2199:and
1635:sinc
1392:sinc
1170:sinc
700:wave
686:The
452:and
411:1909
319:1818
317:and
315:1816
307:1803
291:1675
287:1638
271:1665
246:1803
212:and
84:1660
6366:PMC
6358:doi
6354:369
6303:doi
6291:406
6129:doi
6100:doi
6070:doi
5931::
5805:doi
5716:149
5690::
5643:doi
5592:doi
5536:doi
5441::
5085:sin
4795:or
4661:sin
4647:is
4420:).
4154:sin
4145:sin
4103:cos
4094:sin
4010:sin
3990:cos
3978:sin
3799:cos
3772:sin
3760:sin
3736:cos
3727:sin
2291:),
2094:sin
2074:sin
1848:sin
1827:sin
1682:sin
1670:sin
1407:sin
1350:to
1202:sin
1067:sin
991:min
935:min
924:sin
898:min
855:sin
409:in
348:In
305:in
141:In
6511::
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6364:.
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6325:.
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6301:.
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6096:55
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6056:.
6052:.
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5813:.
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6072::
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5821:.
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5799::
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5742:.
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5586::
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5174:m
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5070:m
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4952:,
4946:p
4943:h
4938:=
4706:D
4686:,
4681:D
4670:=
4639:(
4627:1
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4505:=
4502:x
4483:.
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2368:(
2345:,
2342:)
2338:r
2334:(
2328:=
2320:2
2316:k
2312:+
2304:2
2267:r
2212:1
2208:J
2183:/
2176:2
2156:k
2136:a
2116:,
2111:2
2106:)
2091:a
2088:k
2083:)
2071:a
2068:k
2065:(
2060:1
2056:J
2052:2
2046:(
2039:0
2035:I
2031:=
2028:)
2022:(
2019:I
1947:m
1927:d
1905:i
1880:,
1874:m
1871:=
1867:)
1860:i
1839:m
1823:(
1819:d
1808:m
1804:θ
1745:.
1731:i
1702:]
1698:)
1693:i
1667:(
1655:d
1648:[
1639:2
1628:0
1624:I
1620:=
1617:)
1611:(
1608:I
1582:i
1550:0
1534:,
1516:d
1505:,
1482:,
1464:d
1444:(
1431:,
1417:x
1413:x
1401:=
1398:x
1369:2
1361:=
1336:2
1325:=
1300:0
1297:=
1286:(
1270:0
1266:I
1245:)
1239:(
1236:I
1216:,
1212:)
1190:d
1183:(
1174:2
1163:0
1159:I
1155:=
1152:)
1146:(
1143:I
1112:n
1092:,
1086:n
1083:=
1078:n
1063:d
1041:n
966:d
946:,
940:=
920:d
871:2
867:)
861:(
852:d
838:.
826:2
822:/
789:2
622:.
289:–
285:(
101:d
20:)
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