499:
6423:
1747:
406:; 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).
702:
511:
4863:
750:
6435:
660:
555:
327:
463:
5024:
475:
742:
638:
734:
79:
5236:
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.
618:
1755:
4433:, 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
487:
543:
29:
230:
6471:
668:
1974:
456:, 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.
602:
3055:
3050:
6447:
6483:
4055:
3616:
6459:
4392:
4440:
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
389:
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
829:
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
5235:
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
1985:
765:
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
5250:
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
4874:
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
448:
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
4714:
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
3819:
797:
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
3271:
2774:
5444: : 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.
4771:
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.
5243:. 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
4753:
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.
3398:
2834:
5436: : 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.
1774:
3406:
3824:
5271:. 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.
4811:
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
1215:
4480:. 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
5683:(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:
4912:
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
1739:
5008:
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.
4171:
1790:
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
2115:
3696:
2470:
4847:
5552:
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".
1958:
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.
3136:
498:
3691:
757:
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
770:, 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
1879:
4404:
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
3143:
2646:
4685:
369:
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.
216:
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
4719:
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.
2344:
3276:
1702:
5988:
1778:
5459:
1781:
1780:
1776:
1775:
793:
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.
1782:
4429:
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
1418:
4866:
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.
510:
945:
298:
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.
1127:
530:
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
4723:
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
2586:
4163:
5251:
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.
1091:
5776:
4108:
3045:{\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',}
5846:
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
4885:
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
4835:
872:
4951:
5093:
2003:
1779:
462:
4846:
4516:
2826:
1337:
2388:
1370:
3611:{\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'.}
2375:
5001:
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.
4050:{\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'.}
992:
899:
830:
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
2615:
1732:
1583:
745:
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.
1549:
5035:) 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.
2520:
2186:
1521:
1469:
2637:
2267:
1906:
1042:
5047:. Bragg diffraction is a consequence of interference between waves reflecting from many different crystal planes. The condition of constructive interference is given by
1244:
825:
5475:
Wireless
Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002 pg 126
1299:
5113:
4536:
1803:
1012:
5254:
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.
5194:) 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
5153:
1492:
124:
4626:
158:, 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,
2245:
2213:
1271:
791:
585:- 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
5976:
5212:
5173:
5133:
4995:
4971:
4705:
4600:
4580:
4556:
3065:
2155:
2135:
1946:
1926:
1111:
965:
100:
3623:
2590:
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
2475:
271:
4639:
4387:{\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',}
474:
361:. 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
3814:{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} ,}
258:, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in
2283:
486:
1592:
243:
might have observed diffraction in a broadening of the shadow. The effects of diffraction of light were first carefully observed and characterized by
1777:
4882:
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.
5930:
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.)
4441:
in the vertical direction is higher than in horizontal, the laser beam divergence will be lower in the vertical direction than in the horizontal.
1017:
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
6106:
Kouyoumjian, R. G.; Pathak, P. H. (November 1974). "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface".
6398:
6422:
6250:
6223:
6198:
6171:
6021:
5643:
5496:
4456:
5924:(I have thus shown in what manner one can imagine that light propagates successively by spherical waves, … ) (Note: Huygens published his
904:
4792:
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
5753:
5687:(Proposition 22. Sometimes light, as a result of its transmission, renders dark a body's surface, previously illuminated by another .)
2525:
5913:
5681:"Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte."
5902:
5672:
5423:
4113:
410:
291:
5386:
4889:
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.
567:
This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the
6146:
5536:
4834:
4817:
1047:
4062:
2219:. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
1746:
5685:"Propositio XXII. Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam."
3266:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}},}
2769:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}
4919:
1376:
390:(as opposed to wave properties only arising from the interactions between multitudes of photons) was implied by a low-intensity
5331:
5054:
155:
4483:
6256:
5217:
For completeness, Bragg diffraction is a limit for a large number of atoms with X-rays or neutrons, and is rarely valid for
4793:
759:
581:- a bright disc and rings around a bright light source like the sun or the moon. At the opposite point one may also observe
346:
135:
5267:
A new way to image single biological particles has emerged since the 2010s, utilising the bright X-rays generated by X-ray
1951:
The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a
663:
View from the end of
Millennium Bridge; Moon rising above the Southwark Bridge. Street lights are reflecting in the Thames.
452:
The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For
5922:"J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques, … "
4825:
4401:
3393:{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}
353:. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a
709:
Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.
5316:
5301:
4796:, which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, and creates a new
701:
6508:
2483:
2382:
687:, that is diffraction off the meat fibers. All these effects are a consequence of the fact that light propagates as a
608:
578:
315:
5039:
Diffraction from a large three-dimensional periodic structure such as many thousands of atoms in a crystal is called
6413:
4450:
1423:
713:
656:
in camera or support struts in telescope; In normal vision, diffraction through eyelashes may produce such spikes.
624:
582:
5821:
Fresnel, Augustin-Jean (1816), "Mémoire sur la diffraction de la lumière" ("Memoir on the diffraction of light"),
833:
434:
403:
194:
170:
5247:, the coherence length is related to the lifetime of the excited state from which the atom made its transition.
4862:
679:
which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When
287:
244:
64:
1210:{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}
2781:
6108:
5746:
5286:
4421:
changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially
1306:
1303:
which is also a normalization factor of the intensity profile that can be determined by an integration from
574:
350:
4731:(large aperture diameter compared to working distance) in order to obtain the highest possible resolution.
2349:
1342:
59:
of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the
5341:
5321:
5306:
5182:
Bragg diffraction may be carried out using either electromagnetic radiation of very short wavelength like
4886:
4768:
4763:
1434:
1121:
970:
877:
659:
418:
391:
233:
Thomas Young's sketch of two-slit diffraction for water waves, which he presented to the Royal
Society in
1496:
hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If
5361:
5281:
5183:
4914:
1710:
1561:
1441:
1117:
728:
386:
326:
299:
6434:
554:
1992:
The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the
1528:
749:
6503:
6283:
6050:
5785:
5572:
5381:
5336:
5218:
5191:
4907:
409:
There are various analytical models which allow the diffracted field to be calculated, including the
5825:, vol. 1, pp. 239–81 (March 1816); reprinted as "Deuxième Mémoire…" ("Second Memoir…") in
5457:"A History of Physics in its Elementary Branches, including the evolution of physical laboratories."
5023:
2489:
2160:
1984:
1551:
would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of
634:
A shadow of a solid object, using light from a compact source, shows small fringes near its edges.
5704:
5391:
5268:
5230:
5187:
5044:
4903:
4477:
4422:
4400:
of the aperture distribution. Huygens' principle when applied to an aperture simply says that the
2640:
2593:
2479:
1768:
1586:
1500:
1448:
767:
531:
426:
283:
163:
159:
147:
5728:
5628:
Stationary
Diffraction by Wedges : Method of Automorphic Functions on Complex Characteristics
5456:
2620:
2250:
2110:{\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}
6487:
6475:
6439:
6315:
6270:
Neutze, Richard; Wouts, Remco; van der Spoel, David; Weckert, Edgar; Hajdu, Janos (August 2000).
6076:
5867:
5803:
5604:
5562:
5366:
4821:
4728:
4724:
4716:
4633:
3140:
In the far field, wherein the parallel rays approximation can be employed, the Green's function,
2277:
1884:
1552:
1116:
There is no such simple argument to enable us to find the maxima of the diffraction pattern. The
1020:
649:
430:
422:
311:
186:
182:
5954:
1220:
802:
766:
this system is simplified if we consider light of a single wavelength. If the incident light is
741:
437:. Most configurations cannot be solved analytically, but can yield numerical solutions through
6390:
2478:.) By direct substitution, the solution to this equation can be readily shown to be the scalar
2465:{\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi ).}
1278:
6364:
6307:
6299:
6246:
6219:
6194:
6167:
6142:
6017:
6011:
5926:
5883:
5879:
5863:
5855:
5750:
5639:
5596:
5588:
5532:
5492:
5396:
5311:
5296:
5098:
5040:
5018:
4808:
4521:
4459:
The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this
4397:
4166:
2828:
is incident on the aperture, the field produced by this aperture distribution is given by the
997:
637:
560:
Data is written on CDs as pits and lands; the pits on the surface act as diffracting elements.
374:
338:
282:) observed the diffraction patterns caused by a bird feather, which was effectively the first
240:
213:
and, therefore, undergoes diffraction (which is measurable at subatomic to molecular levels).
206:
131:
6038:
5830:
5138:
1477:
468:
Computer-generated intensity pattern formed on a screen by diffraction from a square aperture
109:
6451:
6354:
6346:
6291:
6117:
6088:
6058:
5834:
5793:
5771:
5669:
5631:
5580:
5524:
5420:
5240:
4813:
4605:
2829:
2378:
2377:
is the 3-dimensional delta function. The delta function has only radial dependence, so the
2270:
733:
705:
Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry
442:
174:
60:
5910:
2230:
2191:
1249:
773:
5984:
5917:
5906:
5757:
5676:
5463:
5427:
5291:
4974:
4746:
4740:
2216:
676:
399:
307:
303:
295:
259:
234:
72:
5698:
6405:
6287:
6054:
5789:
5722:
5576:
78:
51:
is the interference or bending of waves around the corners of an obstacle or through an
6427:
6359:
6334:
6079:(June 2013). "GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations".
5874:, American Book Company, 1900, pp. 81–144. (First published, as extracts only, in
5351:
5346:
5326:
5197:
5158:
5118:
5028:
4980:
4956:
4708:
4690:
4585:
4565:
4541:
4469:
4406:
2140:
2120:
1931:
1911:
1096:
950:
438:
310:, and thereby gave great support to the wave theory of light that had been advanced by
85:
5622:
Komech, Alexander; Merzon, Anatoli (2019), Komech, Alexander; Merzon, Anatoli (eds.),
1754:
150:
source (such as a laser) encounters a slit/aperture that is comparable in size to its
6497:
5807:
5528:
4750:
4461:
4434:
4426:
2273:
414:
362:
358:
4852:
Diffraction on a soft aperture, with a gradient of conductivity over the image width
4455:
617:
6463:
6319:
5812:. (Note: This lecture was presented before the Royal Society on 24 November 1803.)
5608:
5356:
5244:
395:
382:
354:
263:
5896:
3821:
the expression for the
Fraunhofer region field from a planar aperture now becomes
28:
6240:
5486:
1874:{\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}
6384:
5772:"The Bakerian Lecture: Experiments and calculations relative to physical optics"
5635:
5371:
5005:
4899:
4800:. This new wavefront propagates into the geometric shadow area of the obstacle.
4466:
4430:
1952:
586:
542:
331:
229:
210:
202:
6062:
5239:
The length over which the phase in a beam of light is correlated is called the
645:) seen in the center of the shadow of a circular obstacle is due to diffraction
6160:
5741:
Letter from James
Gregory to John Collins, dated 13 May 1673. Reprinted in:
5623:
5376:
4804:
1738:
1706:
The choice of plus/minus sign depends on the definition of the incident angle
684:
667:
642:
590:
568:
517:
218:
151:
20:
6303:
6092:
5592:
4797:
4789:
4476:
The ability of an imaging system to resolve detail is ultimately limited by
1997:
1993:
1968:
1589:), the intensity profile in the Fraunhofer regime (i.e. far field) becomes:
1430:
712:
Diffraction can also be a concern in some technical applications; it sets a
680:
453:
366:
302:
did more definitive studies and calculations of diffraction, made public in
190:
139:
103:
33:
6368:
6350:
6311:
6271:
6121:
5798:
5727:. London: Longman, Rees, Orme, Brown & Green and John Taylor. pp.
5600:
5417:
Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo
146:. The characteristic bending pattern is most pronounced when a wave from a
5584:
3131:{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} .}
1973:
4998:
4629:
4559:
4165:
the Fraunhofer region field of the planar aperture assumes the form of a
3686:{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }
694:
Diffraction can occur with any kind of wave. Ocean waves diffract around
653:
52:
41:
4870:
Several qualitative observations can be made of diffraction in general:
601:
492:
Computational model of an interference pattern from two-slit diffraction
3054:
143:
71:
and was the first to record accurate observations of the phenomenon in
23:, the change in direction of a wave passing from one medium to another.
6339:
Philosophical Transactions of the Royal Society B: Biological Sciences
5221:
or with solid particles in the size range of less than 50 nanometers.
4680:{\displaystyle \theta \approx \sin \theta =1.22{\frac {\lambda }{D}},}
6295:
1585:
of the light onto the slit is non-zero (which causes a change in the
994:
is the angle of incidence at which the minimum intensity occurs, and
378:
56:
6333:
Chapman, Henry N.; Caleman, Carl; Timneanu, Nicusor (17 July 2014).
4816:, who introduced the notion of diffraction coefficients through his
6458:
5666:
Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis …
1758:
A diffraction pattern of a 633 nm laser through a grid of 150 slits
504:
Optical diffraction pattern (laser, analogous to X-ray diffraction)
377:
understanding of light propagation through a slit (or slits) every
40:
beam projected onto a plate after passing through a small circular
6272:"Potential for biomolecular imaging with femtosecond X-ray pulses"
5848:
Mémoires de l'Académie Royale des Sciences de l'Institut de France
5567:
5022:
4861:
4418:
3403:
The expression for the far-zone (Fraunhofer region) field becomes
3053:
2339:{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}
1983:
1972:
1772:
1753:
1745:
1737:
1437:), that is, at a distance much larger than the width of the slit.
748:
740:
732:
700:
695:
666:
658:
636:
534:
to form the familiar rainbow pattern seen when looking at a disc.
325:
252:
228:
198:
178:
154:, as shown in the inserted image. This is due to the addition, or
77:
37:
27:
1988:
Diffraction pattern from a circular aperture at various distances
5909:(Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From
4876:
1697:{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left}
688:
342:
279:
275:
5872:
The Wave Theory of Light: Memoirs by Huygens, Young and Fresnel
5043:. It is similar to what occurs when waves are scattered from a
4396:
In the far-field / Fraunhofer region, this becomes the spatial
480:
Generation of an interference pattern from two-slit diffraction
3062:
where the source point in the aperture is given by the vector
5743:
Correspondence of Scientific Men of the Seventeenth Century …
520:
are partially due to diffraction, according to some analyses.
402:. The quantum approach has some striking similarities to the
5630:, Cham: Springer International Publishing, pp. 15–17,
671:
Simulated diffraction spikes in hexagonal telescope mirrors
421:
approximation of the Kirchhoff equation (applicable to the
6385:
The Feynman Lectures on Physics Vol. I Ch. 30: Diffraction
6218:(4th ed.). United States of America: Addison Wesley.
5519:
Kokkotas, Kostas D. (2003). "Gravitational Wave Physics".
4711:
of the imaging lens (e.g., of a telescope's main mirror).
1413:{\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}
548:
Pixels on smart phone screen acting as diffraction grating
5777:
Philosophical Transactions of the Royal Society of London
716:
to the resolution of a camera, telescope, or microscope.
2227:
The wave that emerges from a point source has amplitude
737:
2D Single-slit diffraction with width changing animation
6158:
Halliday, David; Resnick, Robert; Walker, Jerl (2005),
5700:
Memoires pour l'histoire des sciences et des beaux arts
5668:(Bologna ("Bonomia"), (Italy): Vittorio Bonati, 1665),
4602:) of the imaging optics; this is strictly accurate for
6239:
Ayahiko Ichimiya; Philip I. Cohen (13 December 2004).
1750:
Diffraction of a red laser using a diffraction grating
1742:
2-slit (top) and 5-slit diffraction of red laser light
1345:
1309:
940:{\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}
166:), a complex pattern of varying intensity can result.
6411:
5703:. Paris: Impr. de S. A. S.; Chez E. Ganeau. pp.
5491:. Springer Science & Business Media. p. 14.
5419:(Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665),
5200:
5161:
5141:
5121:
5101:
5057:
4983:
4959:
4922:
4693:
4642:
4608:
4588:
4568:
4544:
4524:
4486:
4174:
4116:
4065:
3827:
3699:
3626:
3409:
3279:
3146:
3068:
2837:
2784:
2649:
2623:
2596:
2528:
2492:
2391:
2352:
2286:
2253:
2233:
2194:
2163:
2143:
2123:
2006:
1934:
1914:
1887:
1806:
1713:
1595:
1564:
1531:
1503:
1480:
1451:
1379:
1281:
1252:
1223:
1130:
1099:
1050:
1023:
1000:
973:
953:
907:
880:
836:
805:
776:
112:
88:
5868:"Fresnel's prize memoir on the diffraction of light"
5485:
Suryanarayana, C.; Norton, M. Grant (29 June 2013).
4824:
extended the (singular) Keller coefficients via the
719:
Other examples of diffraction are considered below.
652:
are diffraction patterns caused due to non-circular
5977:"Food Explainer: Why Is Some Deli Meat Iridescent?"
5862:, vol. 1 (Paris: Imprimerie Impériale, 1866),
5829:, vol. 1 (Paris: Imprimerie Impériale, 1866),
2581:{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}
6159:
5206:
5167:
5147:
5127:
5107:
5087:
4989:
4965:
4945:
4699:
4679:
4620:
4594:
4574:
4550:
4530:
4510:
4386:
4157:
4102:
4049:
3813:
3685:
3610:
3392:
3265:
3130:
3044:
2820:
2768:
2631:
2609:
2580:
2514:
2464:
2369:
2338:
2261:
2239:
2207:
2180:
2149:
2129:
2109:
1940:
1920:
1900:
1873:
1726:
1696:
1577:
1543:
1515:
1486:
1463:
1412:
1364:
1331:
1293:
1265:
1238:
1209:
1105:
1085:
1036:
1006:
986:
959:
939:
893:
866:
819:
785:
118:
94:
82:Infinitely many points (three shown) along length
6037:Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964).
4255:
4254:
3908:
3907:
3802:
3778:
3739:
3706:
3677:
3654:
3570:
3490:
3489:
3379:
3119:
3096:
2887:
2886:
1948:is an integer which can be positive or negative.
874:so that the minimum intensity occurs at an angle
134:, the diffraction phenomenon is described by the
6166:(7th ed.), USA: John Wiley and Sons, Inc.,
5214:, allowing one to deduce the crystal structure.
4158:{\displaystyle k_{y}=k\sin \theta \sin \phi \,,}
627:, as seen from a plane on the underlying clouds.
5521:Encyclopedia of Physical Science and Technology
4803:Knife-edge diffraction is an outgrowth of the "
5745:, ed. Stephen Jordan Rigaud (Oxford, England:
3058:On the calculation of Fraunhofer region fields
1086:{\displaystyle d\,\sin \theta _{n}=n\lambda ,}
6184:
6182:
5941:The Mathematical Theory of Huygens' Principle
4788:is a truncation of a portion of the incident
4103:{\displaystyle k_{x}=k\sin \theta \cos \phi }
1908:is the angle at which the light is incident,
1473:the intensity will have little dependency on
314:and reinvigorated by Young, against Newton's
266:studied these effects and attributed them to
106:, producing a continuously varying intensity
8:
2617:and the field point is located at the point
2476:del in cylindrical and spherical coordinates
593:and internal reflection within the droplet.
6242:Reflection High-Energy Electron Diffraction
4632:case). In object space, the corresponding
1928:is the separation of grating elements, and
867:{\displaystyle {\frac {d\sin(\theta )}{2}}}
589:of the light that forms the glory involves
4946:{\displaystyle \lambda ={\frac {h}{p}}\,,}
1955:of diffraction and interference patterns.
753:Graph and image of single-slit diffraction
6358:
5797:
5749:, 1841), vol. 2, pp. 251–255, especially
5566:
5199:
5160:
5155:is the angle of the diffracted wave, and
5140:
5120:
5100:
5088:{\displaystyle m\lambda =2d\sin \theta ,}
5056:
4982:
4958:
4939:
4929:
4921:
4692:
4664:
4641:
4607:
4587:
4567:
4543:
4523:
4485:
4369:
4357:
4338:
4317:
4303:
4261:
4260:
4226:
4225:
4196:
4190:
4173:
4151:
4121:
4115:
4070:
4064:
4032:
4020:
3956:
3914:
3913:
3879:
3878:
3849:
3843:
3826:
3797:
3796:
3773:
3772:
3734:
3733:
3701:
3700:
3698:
3672:
3671:
3649:
3648:
3628:
3625:
3593:
3581:
3565:
3564:
3552:
3538:
3496:
3495:
3461:
3460:
3431:
3425:
3408:
3374:
3373:
3361:
3347:
3318:
3312:
3297:
3291:
3286:
3278:
3252:
3243:
3234:
3229:
3215:
3206:
3197:
3192:
3185:
3179:
3164:
3158:
3153:
3145:
3114:
3113:
3091:
3090:
3070:
3067:
3027:
3015:
3007:
2998:
2989:
2984:
2970:
2961:
2952:
2947:
2940:
2934:
2893:
2892:
2858:
2857:
2836:
2790:
2789:
2783:
2755:
2746:
2737:
2732:
2718:
2709:
2700:
2695:
2688:
2682:
2667:
2661:
2656:
2648:
2624:
2622:
2598:
2595:
2550:
2544:
2527:
2497:
2491:
2438:
2424:
2418:
2408:
2396:
2390:
2359:
2351:
2325:
2307:
2291:
2285:
2254:
2252:
2232:
2199:
2193:
2170:
2162:
2142:
2122:
2098:
2047:
2037:
2026:
2005:
1933:
1913:
1892:
1886:
1847:
1842:
1826:
1821:
1805:
1718:
1712:
1680:
1640:
1626:
1621:
1615:
1594:
1569:
1563:
1530:
1502:
1479:
1450:
1392:
1378:
1352:
1344:
1319:
1308:
1280:
1257:
1251:
1222:
1175:
1161:
1156:
1150:
1129:
1098:
1065:
1054:
1049:
1028:
1022:
999:
978:
972:
952:
922:
911:
906:
885:
879:
837:
835:
809:
804:
775:
111:
87:
5135:is the distance between crystal planes,
4454:
2137:is the radius of the circular aperture,
1800:which are given by the grating equation
1273:is the intensity at the central maximum
209:also demonstrates that matter possesses
173:travels through a medium with a varying
142:as a collection of individual spherical
138:that treats each point in a propagating
6418:
5939:Baker, B.B. & Copson, E.T. (1939),
5488:X-Ray Diffraction: A Practical Approach
5408:
4830:
4511:{\displaystyle \Delta x=1.22\lambda N,}
4417:The way in which the beam profile of a
3400:as can be seen in the adjacent figure.
2821:{\displaystyle E_{\mathrm {inc} }(x,y)}
2486:(and using the physics time convention
1332:{\textstyle \theta =-{\frac {\pi }{2}}}
458:
6399:International Union of Crystallography
6133:
6131:
6081:IEEE Antennas and Propagation Magazine
5991:from the original on 10 September 2013
1365:{\textstyle \theta ={\frac {\pi }{2}}}
330:Single-slit diffraction in a circular
181:travels through a medium with varying
5854:(for 1821 & 1822, printed 1826),
4820:(GTD). In 1974, Prabhakar Pathak and
2370:{\displaystyle \delta (\mathbf {r} )}
2269:that is given by the solution of the
102:project phase contributions from the
7:
5975:Arumugam, Nadia (9 September 2013).
5860:Oeuvres complètes d'Augustin Fresnel
5827:Oeuvres complètes d'Augustin Fresnel
4840:Diffraction on a sharp metallic edge
2000:in intensity with angle is given by
987:{\displaystyle \theta _{\text{min}}}
894:{\displaystyle \theta _{\text{min}}}
345:propagate; this is described by the
6406:Using a cd as a diffraction grating
6013:Dynamic fields and waves of physics
5837:submitted on 15 October 1815.)
4538:is the wavelength of the light and
2643:(for arbitrary source location) as
2639:, then we may represent the scalar
1246:is the intensity at a given angle,
351:principle of superposition of waves
6259:from the original on 16 July 2017.
4487:
4268:
4265:
4262:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4175:
3921:
3918:
3915:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3828:
3503:
3500:
3497:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3410:
2900:
2897:
2894:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2838:
2797:
2794:
2791:
2431:
2421:
2393:
2288:
1727:{\displaystyle \theta _{\text{i}}}
1578:{\displaystyle \theta _{\text{i}}}
1429:This analysis applies only to the
429:approximation (applicable to the
341:diffraction arises because of how
14:
5624:"The Early Theory of Diffraction"
4818:geometrical theory of diffraction
2381:(a.k.a. scalar Laplacian) in the
1977:A computer-generated image of an
16:Phenomenon of the motion of waves
6481:
6469:
6457:
6445:
6433:
6421:
6335:"Diffraction before destruction"
6039:"Self-Trapping of Optical Beams"
5955:"Optical effects on spider webs"
5878:, vol. 11 (1819), pp.
5876:Annales de Chimie et de Physique
5823:Annales de Chimie et de Physique
5466:MacMillan Company, New York 1899
4845:
4833:
3799:
3775:
3736:
3703:
3674:
3651:
3629:
3567:
3553:
3376:
3362:
3298:
3287:
3244:
3235:
3207:
3198:
3165:
3154:
3116:
3093:
3071:
2999:
2990:
2962:
2953:
2778:Therefore, if an electric field
2747:
2738:
2710:
2701:
2668:
2657:
2625:
2599:
2360:
2326:
2255:
1544:{\displaystyle \theta \approx 0}
1372:and conservation of energy, and
1014:is the wavelength of the light.
616:
600:
571:on a credit card is an example.
553:
541:
509:
497:
485:
473:
461:
185:– all waves diffract, including
169:These effects also occur when a
5770:Thomas Young (1 January 1804).
5332:Dynamical theory of diffraction
4807:problem", originally solved by
1113:is an integer other than zero.
577:by small particles can cause a
55:into the region of geometrical
6245:. Cambridge University Press.
5529:10.1016/B0-12-227410-5/00300-8
5263:Diffraction before destruction
4352:
4310:
4296:
4274:
4184:
4178:
4015:
3975:
3949:
3927:
3837:
3831:
3576:
3548:
3531:
3509:
3419:
3413:
3385:
3357:
3306:
3292:
3283:
3253:
3230:
3216:
3193:
3173:
3159:
3150:
3008:
2985:
2971:
2948:
2928:
2906:
2847:
2841:
2815:
2803:
2756:
2733:
2719:
2696:
2676:
2662:
2653:
2538:
2532:
2515:{\displaystyle e^{-i\omega t}}
2456:
2447:
2364:
2356:
2330:
2322:
2181:{\displaystyle 2\pi /\lambda }
2071:
2053:
2016:
2010:
1686:
1655:
1605:
1599:
1233:
1227:
1140:
1134:
855:
849:
411:Kirchhoff diffraction equation
1:
6191:Introduction to Modern Optics
5387:Schaefer–Bergmann diffraction
4826:uniform theory of diffraction
4582:divided by aperture diameter
4402:far-field diffraction pattern
2610:{\displaystyle \mathbf {r} '}
1516:{\displaystyle d\gg \lambda }
1464:{\displaystyle d\ll \lambda }
575:Diffraction in the atmosphere
6391:"Scattering and diffraction"
5709:grimaldi diffraction 0–1800.
5317:Diffraction vs. interference
5302:Coherent diffraction imaging
4425:wave front, it approximates
2632:{\displaystyle \mathbf {r} }
2262:{\displaystyle \mathbf {r} }
1120:can be calculated using the
6141:(North-Holland, Amsterdam)
5721:Sir David Brewster (1831).
5636:10.1007/978-3-030-26699-8_2
5175:is an integer known as the
4445:Diffraction-limited imaging
4413:Propagation of a laser beam
2484:spherical coordinate system
2383:spherical coordinate system
1901:{\displaystyle \theta _{i}}
1037:{\displaystyle \theta _{n}}
316:corpuscular theory of light
247:, who also coined the term
6525:
6063:10.1103/PhysRevLett.13.479
6016:. CRC Press. p. 102.
5943:, Oxford, pp. 36–40.
5870:, in H. Crew (ed.),
5697:Jean Louis Aubert (1760).
5664:Francesco Maria Grimaldi,
5415:Francesco Maria Grimaldi,
5228:
5016:
4897:
4761:
4738:
4451:Diffraction-limited system
4448:
1966:
1766:
1424:unnormalized sinc function
1239:{\displaystyle I(\theta )}
967:is the width of the slit,
820:{\displaystyle \lambda /2}
726:
18:
4794:Huygens–Fresnel principle
4757:
1294:{\displaystyle \theta =0}
760:Huygens–Fresnel principle
435:path integral formulation
404:Huygens-Fresnel principle
375:modern quantum mechanical
347:Huygens–Fresnel principle
136:Huygens–Fresnel principle
6189:Grant R. Fowles (1975).
6093:10.1109/MAP.2013.6586622
5179:of the diffracted beam.
5108:{\displaystyle \lambda }
4531:{\displaystyle \lambda }
2276:for a point source (the
1558:When the incident angle
1007:{\displaystyle \lambda }
387:probability distribution
245:Francesco Maria Grimaldi
126:on the registering plate
65:Francesco Maria Grimaldi
63:wave. Italian scientist
19:Not to be confused with
6193:. Courier Corporation.
6109:Proceedings of the IEEE
6043:Physical Review Letters
5866:; partly translated as
5747:Oxford University Press
5287:Atmospheric diffraction
5148:{\displaystyle \theta }
4894:Matter wave diffraction
4707:is the diameter of the
1487:{\displaystyle \theta }
723:Single-slit diffraction
160:closely spaced openings
119:{\displaystyle \theta }
6351:10.1098/rstb.2013.0313
6214:Hecht, Eugene (2002).
6162:Fundamental of Physics
6137:John M. Cowley (1975)
6122:10.1109/PROC.1974.9651
6010:Andrew Norton (2000).
5799:10.1098/rstl.1804.0001
5446:
5438:
5342:Fraunhofer diffraction
5322:Diffractive solar sail
5307:Diffraction from slits
5208:
5169:
5149:
5129:
5109:
5089:
5036:
4991:
4967:
4947:
4867:
4786:knife-edge diffraction
4701:
4681:
4622:
4621:{\displaystyle N\gg 1}
4596:
4576:
4552:
4532:
4512:
4473:
4388:
4159:
4104:
4051:
3815:
3687:
3612:
3394:
3267:
3132:
3059:
3046:
2822:
2770:
2633:
2611:
2582:
2516:
2466:
2371:
2340:
2263:
2241:
2209:
2182:
2151:
2131:
2111:
1989:
1981:
1942:
1922:
1902:
1875:
1787:
1759:
1751:
1743:
1728:
1698:
1579:
1545:
1517:
1488:
1465:
1435:Fraunhofer diffraction
1414:
1366:
1333:
1295:
1267:
1240:
1211:
1122:Fraunhofer diffraction
1107:
1087:
1038:
1008:
988:
961:
941:
895:
868:
821:
787:
754:
746:
738:
706:
672:
664:
646:
419:Fraunhofer diffraction
392:double-slit experiment
334:
237:
127:
120:
96:
45:
5585:10.1038/nnano.2012.34
5555:Nature Nanotechnology
5439:
5431:
5362:Point spread function
5282:Angle-sensitive pixel
5209:
5186:or matter waves like
5170:
5150:
5130:
5110:
5090:
5026:
5017:Further information:
4992:
4968:
4948:
4915:de Broglie wavelength
4865:
4725:microscope objectives
4702:
4682:
4623:
4597:
4577:
4553:
4533:
4513:
4458:
4389:
4160:
4105:
4052:
3816:
3688:
3613:
3395:
3268:
3133:
3057:
3047:
2823:
2771:
2634:
2612:
2583:
2517:
2467:
2372:
2341:
2264:
2242:
2240:{\displaystyle \psi }
2210:
2208:{\displaystyle J_{1}}
2183:
2152:
2132:
2112:
1987:
1976:
1943:
1923:
1903:
1876:
1785:
1757:
1749:
1741:
1729:
1699:
1580:
1546:
1518:
1489:
1466:
1415:
1367:
1334:
1296:
1268:
1266:{\displaystyle I_{0}}
1241:
1212:
1108:
1088:
1039:
1009:
989:
962:
942:
896:
869:
822:
788:
786:{\displaystyle 2\pi }
752:
744:
736:
729:Diffraction formalism
704:
698:and other obstacles.
670:
662:
640:
329:
300:Augustin-Jean Fresnel
292:celebrated experiment
232:
195:electromagnetic waves
121:
97:
81:
31:
5898:Traité de la lumiere
5895:Christiaan Huygens,
5724:A Treatise on Optics
5337:Electron diffraction
5269:free-electron lasers
5219:electron diffraction
5198:
5159:
5139:
5119:
5099:
5055:
4981:
4957:
4920:
4908:Electron diffraction
4691:
4640:
4606:
4586:
4566:
4542:
4522:
4484:
4172:
4114:
4063:
3825:
3697:
3624:
3407:
3277:
3144:
3066:
2835:
2782:
2647:
2621:
2594:
2526:
2490:
2389:
2350:
2284:
2251:
2231:
2192:
2161:
2141:
2121:
2004:
1932:
1912:
1885:
1804:
1711:
1593:
1562:
1529:
1501:
1478:
1449:
1377:
1343:
1307:
1279:
1250:
1221:
1128:
1097:
1048:
1021:
998:
971:
951:
905:
878:
834:
803:
774:
385:that determines the
381:is described by its
211:wave-like properties
110:
86:
6288:2000Natur.406..752N
6139:Diffraction physics
6077:Rahmat-Samii, Yahya
6055:1964PhRvL..13..479C
5953:Dietrich Zawischa.
5835:"First Memoir"
5833:. (Revision of the
5790:1804RSPT...94....1Y
5577:2012NatNa...7..297J
5392:Thinned-array curse
5231:Coherence (physics)
5115:is the wavelength,
5045:diffraction grating
4904:Neutron diffraction
4769:Babinet's principle
4764:Babinet's principle
4758:Babinet's principle
1786:Diffraction grating
1769:Diffraction grating
1763:Diffraction grating
532:diffraction grating
427:Fresnel diffraction
394:first performed by
284:diffraction grating
187:gravitational waves
164:diffraction grating
34:diffraction pattern
6509:Physical phenomena
6345:(1647): 20130313.
5916:2016-12-01 at the
5905:2016-06-16 at the
5756:2016-12-01 at the
5675:2016-12-01 at the
5462:2016-12-01 at the
5426:2016-12-01 at the
5367:Powder diffraction
5204:
5165:
5145:
5125:
5105:
5085:
5037:
4987:
4963:
4943:
4868:
4822:Robert Kouyoumjian
4749:seen when using a
4729:numerical aperture
4717:Rayleigh criterion
4697:
4677:
4634:angular resolution
4618:
4592:
4572:
4548:
4528:
4508:
4474:
4384:
4253:
4155:
4100:
4047:
3906:
3811:
3683:
3608:
3488:
3390:
3263:
3128:
3060:
3042:
2885:
2818:
2766:
2629:
2607:
2578:
2512:
2462:
2367:
2336:
2278:Helmholtz equation
2259:
2237:
2205:
2178:
2147:
2127:
2107:
1990:
1982:
1938:
1918:
1898:
1871:
1788:
1760:
1752:
1744:
1724:
1694:
1575:
1553:geometrical optics
1541:
1513:
1484:
1461:
1410:
1362:
1329:
1291:
1263:
1236:
1207:
1103:
1083:
1034:
1004:
984:
957:
937:
891:
864:
817:
783:
755:
747:
739:
707:
673:
665:
650:Diffraction spikes
647:
433:) and the Feynman
413:(derived from the
335:
312:Christiaan Huygens
286:to be discovered.
238:
183:acoustic impedance
128:
116:
92:
46:
6282:(6797): 752–757.
6252:978-0-521-45373-8
6225:978-0-8053-8566-3
6200:978-0-486-65957-2
6173:978-0-471-23231-5
6116:(11): 1448–1461.
6023:978-0-7503-0719-2
5645:978-3-030-26699-8
5498:978-1-4899-0148-4
5397:X-ray diffraction
5312:Diffraction spike
5297:Cloud iridescence
5245:atomic transition
5207:{\displaystyle d}
5168:{\displaystyle m}
5128:{\displaystyle d}
5041:Bragg diffraction
5019:Bragg diffraction
5013:Bragg diffraction
4990:{\displaystyle p}
4966:{\displaystyle h}
4937:
4809:Arnold Sommerfeld
4782:knife-edge effect
4700:{\displaystyle D}
4672:
4595:{\displaystyle D}
4575:{\displaystyle f}
4551:{\displaystyle N}
4398:Fourier transform
4221:
4219:
4167:Fourier transform
3874:
3872:
3805:
3781:
3771:
3759:
3742:
3709:
3680:
3657:
3573:
3456:
3454:
3382:
3341:
3258:
3122:
3099:
3013:
2933:
2853:
2761:
2573:
2445:
2416:
2150:{\displaystyle k}
2130:{\displaystyle a}
2092:
1963:Circular aperture
1941:{\displaystyle m}
1921:{\displaystyle d}
1783:
1721:
1683:
1653:
1572:
1442:intensity profile
1408:
1360:
1327:
1188:
1118:intensity profile
1106:{\displaystyle n}
981:
960:{\displaystyle d}
925:
888:
862:
714:fundamental limit
641:The bright spot (
516:Colors seen in a
339:classical physics
207:quantum mechanics
132:classical physics
95:{\displaystyle d}
6516:
6486:
6485:
6484:
6474:
6473:
6472:
6462:
6461:
6450:
6449:
6448:
6438:
6437:
6426:
6425:
6417:
6402:
6373:
6372:
6362:
6330:
6324:
6323:
6296:10.1038/35021099
6267:
6261:
6260:
6236:
6230:
6229:
6211:
6205:
6204:
6186:
6177:
6176:
6165:
6155:
6149:
6135:
6126:
6125:
6103:
6097:
6096:
6073:
6067:
6066:
6034:
6028:
6027:
6007:
6001:
6000:
5998:
5996:
5972:
5966:
5965:
5963:
5961:
5950:
5944:
5937:
5931:
5893:
5887:
5864:pp. 247–364
5856:pp. 339–475
5853:
5844:
5838:
5819:
5813:
5811:
5801:
5767:
5761:
5739:
5733:
5732:
5718:
5712:
5711:
5694:
5688:
5662:
5656:
5655:
5654:
5652:
5619:
5613:
5612:
5570:
5549:
5543:
5542:
5516:
5510:
5509:
5507:
5505:
5482:
5476:
5473:
5467:
5455:Cajori, Florian
5453:
5447:
5413:
5241:coherence length
5213:
5211:
5210:
5205:
5174:
5172:
5171:
5166:
5154:
5152:
5151:
5146:
5134:
5132:
5131:
5126:
5114:
5112:
5111:
5106:
5094:
5092:
5091:
5086:
4996:
4994:
4993:
4988:
4972:
4970:
4969:
4964:
4952:
4950:
4949:
4944:
4938:
4930:
4849:
4837:
4814:Joseph B. Keller
4735:Speckle patterns
4727:require a large
4706:
4704:
4703:
4698:
4686:
4684:
4683:
4678:
4673:
4665:
4627:
4625:
4624:
4619:
4601:
4599:
4598:
4593:
4581:
4579:
4578:
4573:
4557:
4555:
4554:
4549:
4537:
4535:
4534:
4529:
4517:
4515:
4514:
4509:
4393:
4391:
4390:
4385:
4380:
4368:
4356:
4355:
4351:
4343:
4342:
4330:
4322:
4321:
4295:
4284:
4273:
4272:
4271:
4252:
4251:
4220:
4218:
4207:
4206:
4191:
4164:
4162:
4161:
4156:
4126:
4125:
4109:
4107:
4106:
4101:
4075:
4074:
4056:
4054:
4053:
4048:
4043:
4031:
4019:
4018:
4014:
3994:
3948:
3937:
3926:
3925:
3924:
3905:
3904:
3873:
3871:
3860:
3859:
3844:
3820:
3818:
3817:
3812:
3807:
3806:
3798:
3783:
3782:
3774:
3769:
3757:
3744:
3743:
3735:
3711:
3710:
3702:
3692:
3690:
3689:
3684:
3682:
3681:
3673:
3670:
3659:
3658:
3650:
3647:
3636:
3632:
3617:
3615:
3614:
3609:
3604:
3592:
3580:
3579:
3575:
3574:
3566:
3560:
3556:
3530:
3519:
3508:
3507:
3506:
3487:
3486:
3455:
3453:
3442:
3441:
3426:
3399:
3397:
3396:
3391:
3389:
3388:
3384:
3383:
3375:
3369:
3365:
3342:
3340:
3329:
3328:
3313:
3305:
3301:
3295:
3290:
3272:
3270:
3269:
3264:
3259:
3257:
3256:
3251:
3247:
3238:
3233:
3221:
3220:
3219:
3214:
3210:
3201:
3196:
3180:
3172:
3168:
3162:
3157:
3137:
3135:
3134:
3129:
3124:
3123:
3115:
3112:
3101:
3100:
3092:
3089:
3078:
3074:
3051:
3049:
3048:
3043:
3038:
3026:
3014:
3012:
3011:
3006:
3002:
2993:
2988:
2976:
2975:
2974:
2969:
2965:
2956:
2951:
2935:
2931:
2927:
2916:
2905:
2904:
2903:
2884:
2883:
2830:surface integral
2827:
2825:
2824:
2819:
2802:
2801:
2800:
2775:
2773:
2772:
2767:
2762:
2760:
2759:
2754:
2750:
2741:
2736:
2724:
2723:
2722:
2717:
2713:
2704:
2699:
2683:
2675:
2671:
2665:
2660:
2641:Green's function
2638:
2636:
2635:
2630:
2628:
2616:
2614:
2613:
2608:
2606:
2602:
2587:
2585:
2584:
2579:
2574:
2572:
2561:
2560:
2545:
2521:
2519:
2518:
2513:
2511:
2510:
2480:Green's function
2471:
2469:
2468:
2463:
2446:
2444:
2443:
2442:
2429:
2428:
2419:
2417:
2409:
2401:
2400:
2379:Laplace operator
2376:
2374:
2373:
2368:
2363:
2345:
2343:
2342:
2337:
2329:
2312:
2311:
2296:
2295:
2271:frequency domain
2268:
2266:
2265:
2260:
2258:
2246:
2244:
2243:
2238:
2223:General aperture
2214:
2212:
2211:
2206:
2204:
2203:
2187:
2185:
2184:
2179:
2174:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2116:
2114:
2113:
2108:
2103:
2102:
2097:
2093:
2091:
2074:
2052:
2051:
2038:
2031:
2030:
1947:
1945:
1944:
1939:
1927:
1925:
1924:
1919:
1907:
1905:
1904:
1899:
1897:
1896:
1880:
1878:
1877:
1872:
1858:
1854:
1853:
1852:
1851:
1832:
1831:
1830:
1784:
1735:
1733:
1731:
1730:
1725:
1723:
1722:
1719:
1703:
1701:
1700:
1695:
1693:
1689:
1685:
1684:
1681:
1654:
1649:
1641:
1631:
1630:
1620:
1619:
1584:
1582:
1581:
1576:
1574:
1573:
1570:
1550:
1548:
1547:
1542:
1524:
1522:
1520:
1519:
1514:
1495:
1493:
1491:
1490:
1485:
1472:
1470:
1468:
1467:
1462:
1421:
1419:
1417:
1416:
1411:
1409:
1404:
1393:
1371:
1369:
1368:
1363:
1361:
1353:
1338:
1336:
1335:
1330:
1328:
1320:
1302:
1300:
1298:
1297:
1292:
1272:
1270:
1269:
1264:
1262:
1261:
1245:
1243:
1242:
1237:
1216:
1214:
1213:
1208:
1203:
1199:
1189:
1184:
1176:
1166:
1165:
1155:
1154:
1112:
1110:
1109:
1104:
1092:
1090:
1089:
1084:
1070:
1069:
1043:
1041:
1040:
1035:
1033:
1032:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
983:
982:
979:
966:
964:
963:
958:
946:
944:
943:
938:
927:
926:
923:
900:
898:
897:
892:
890:
889:
886:
873:
871:
870:
865:
863:
858:
838:
828:
826:
824:
823:
818:
813:
792:
790:
789:
784:
620:
604:
557:
545:
513:
501:
489:
477:
465:
443:boundary element
357:for a secondary
175:refractive index
125:
123:
122:
117:
101:
99:
98:
93:
67:coined the word
44:in another plate
6524:
6523:
6519:
6518:
6517:
6515:
6514:
6513:
6494:
6493:
6492:
6482:
6480:
6470:
6468:
6456:
6446:
6444:
6432:
6420:
6412:
6395:Crystallography
6389:
6381:
6376:
6332:
6331:
6327:
6269:
6268:
6264:
6253:
6238:
6237:
6233:
6226:
6213:
6212:
6208:
6201:
6188:
6187:
6180:
6174:
6157:
6156:
6152:
6136:
6129:
6105:
6104:
6100:
6075:
6074:
6070:
6049:(15): 479–482.
6036:
6035:
6031:
6024:
6009:
6008:
6004:
5994:
5992:
5985:The Slate Group
5974:
5973:
5969:
5959:
5957:
5952:
5951:
5947:
5938:
5934:
5918:Wayback Machine
5907:Wayback Machine
5894:
5890:
5858:; reprinted in
5851:
5845:
5841:
5831:pp. 89–122
5820:
5816:
5769:
5768:
5764:
5758:Wayback Machine
5740:
5736:
5720:
5719:
5715:
5696:
5695:
5691:
5677:Wayback Machine
5663:
5659:
5650:
5648:
5646:
5621:
5620:
5616:
5551:
5550:
5546:
5539:
5518:
5517:
5513:
5503:
5501:
5499:
5484:
5483:
5479:
5474:
5470:
5464:Wayback Machine
5454:
5450:
5428:Wayback Machine
5414:
5410:
5406:
5401:
5292:Brocken spectre
5277:
5265:
5260:
5233:
5227:
5196:
5195:
5157:
5156:
5137:
5136:
5117:
5116:
5097:
5096:
5053:
5052:
5031:, each dot (or
5021:
5015:
5004:Diffraction of
4979:
4978:
4975:Planck constant
4955:
4954:
4918:
4917:
4910:
4896:
4879:of the angles.)
4860:
4853:
4850:
4841:
4838:
4778:
4766:
4760:
4747:speckle pattern
4743:
4741:Speckle pattern
4737:
4689:
4688:
4638:
4637:
4604:
4603:
4584:
4583:
4564:
4563:
4540:
4539:
4520:
4519:
4482:
4481:
4453:
4447:
4415:
4373:
4361:
4344:
4334:
4323:
4313:
4299:
4288:
4277:
4256:
4208:
4192:
4170:
4169:
4117:
4112:
4111:
4066:
4061:
4060:
4036:
4024:
4007:
3987:
3952:
3941:
3930:
3909:
3861:
3845:
3823:
3822:
3695:
3694:
3663:
3640:
3627:
3622:
3621:
3597:
3585:
3551:
3534:
3523:
3512:
3491:
3443:
3427:
3405:
3404:
3360:
3343:
3330:
3314:
3296:
3275:
3274:
3242:
3222:
3205:
3181:
3163:
3142:
3141:
3105:
3082:
3069:
3064:
3063:
3031:
3019:
2997:
2977:
2960:
2936:
2920:
2909:
2888:
2833:
2832:
2785:
2780:
2779:
2745:
2725:
2708:
2684:
2666:
2645:
2644:
2619:
2618:
2597:
2592:
2591:
2562:
2546:
2524:
2523:
2493:
2488:
2487:
2482:, which in the
2434:
2430:
2420:
2392:
2387:
2386:
2348:
2347:
2303:
2287:
2282:
2281:
2249:
2248:
2229:
2228:
2225:
2217:Bessel function
2195:
2190:
2189:
2159:
2158:
2139:
2138:
2119:
2118:
2075:
2043:
2039:
2033:
2032:
2022:
2002:
2001:
1971:
1965:
1930:
1929:
1910:
1909:
1888:
1883:
1882:
1843:
1822:
1814:
1810:
1802:
1801:
1799:
1773:
1771:
1765:
1714:
1709:
1708:
1707:
1676:
1642:
1639:
1635:
1622:
1611:
1591:
1590:
1565:
1560:
1559:
1527:
1526:
1499:
1498:
1497:
1476:
1475:
1474:
1447:
1446:
1445:
1394:
1375:
1374:
1373:
1341:
1340:
1305:
1304:
1277:
1276:
1274:
1253:
1248:
1247:
1219:
1218:
1177:
1174:
1170:
1157:
1146:
1126:
1125:
1095:
1094:
1061:
1046:
1045:
1024:
1019:
1018:
996:
995:
974:
969:
968:
949:
948:
918:
903:
902:
881:
876:
875:
839:
832:
831:
801:
800:
799:
772:
771:
731:
725:
677:speckle pattern
632:
631:
630:
629:
628:
621:
613:
612:
605:
565:
564:
563:
562:
561:
558:
550:
549:
546:
528:
521:
514:
505:
502:
493:
490:
481:
478:
469:
466:
365:as well as the
324:
270:of light rays.
227:
205:. Furthermore,
108:
107:
84:
83:
24:
17:
12:
11:
5:
6522:
6520:
6512:
6511:
6506:
6496:
6495:
6491:
6490:
6478:
6466:
6454:
6442:
6430:
6410:
6409:
6403:
6387:
6380:
6379:External links
6377:
6375:
6374:
6325:
6262:
6251:
6231:
6224:
6206:
6199:
6178:
6172:
6150:
6127:
6098:
6068:
6029:
6022:
6002:
5967:
5945:
5932:
5888:
5839:
5814:
5762:
5734:
5713:
5689:
5657:
5644:
5614:
5561:(5): 297–300.
5544:
5537:
5511:
5497:
5477:
5468:
5448:
5407:
5405:
5402:
5400:
5399:
5394:
5389:
5384:
5379:
5374:
5369:
5364:
5359:
5354:
5352:Fresnel number
5349:
5347:Fresnel imager
5344:
5339:
5334:
5329:
5327:Diffractometer
5324:
5319:
5314:
5309:
5304:
5299:
5294:
5289:
5284:
5278:
5276:
5273:
5264:
5261:
5259:
5256:
5229:Main article:
5226:
5223:
5203:
5164:
5144:
5124:
5104:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5014:
5011:
4986:
4962:
4942:
4936:
4933:
4928:
4925:
4895:
4892:
4891:
4890:
4883:
4880:
4859:
4856:
4855:
4854:
4851:
4844:
4842:
4839:
4832:
4777:
4774:
4762:Main article:
4759:
4756:
4739:Main article:
4736:
4733:
4709:entrance pupil
4696:
4676:
4671:
4668:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4617:
4614:
4611:
4591:
4571:
4562:(focal length
4547:
4527:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4449:Main article:
4446:
4443:
4414:
4411:
4407:Fourier optics
4383:
4379:
4376:
4372:
4367:
4364:
4360:
4354:
4350:
4347:
4341:
4337:
4333:
4329:
4326:
4320:
4316:
4312:
4309:
4306:
4302:
4298:
4294:
4291:
4287:
4283:
4280:
4276:
4270:
4267:
4264:
4259:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4224:
4217:
4214:
4211:
4205:
4202:
4199:
4195:
4189:
4186:
4183:
4180:
4177:
4154:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4124:
4120:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4073:
4069:
4046:
4042:
4039:
4035:
4030:
4027:
4023:
4017:
4013:
4010:
4006:
4003:
4000:
3997:
3993:
3990:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3955:
3951:
3947:
3944:
3940:
3936:
3933:
3929:
3923:
3920:
3917:
3912:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3877:
3870:
3867:
3864:
3858:
3855:
3852:
3848:
3842:
3839:
3836:
3833:
3830:
3810:
3804:
3801:
3795:
3792:
3789:
3786:
3780:
3777:
3768:
3765:
3762:
3756:
3753:
3750:
3747:
3741:
3738:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3708:
3705:
3679:
3676:
3669:
3666:
3662:
3656:
3653:
3646:
3643:
3639:
3635:
3631:
3607:
3603:
3600:
3596:
3591:
3588:
3584:
3578:
3572:
3569:
3563:
3559:
3555:
3550:
3547:
3544:
3541:
3537:
3533:
3529:
3526:
3522:
3518:
3515:
3511:
3505:
3502:
3499:
3494:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3459:
3452:
3449:
3446:
3440:
3437:
3434:
3430:
3424:
3421:
3418:
3415:
3412:
3387:
3381:
3378:
3372:
3368:
3364:
3359:
3356:
3353:
3350:
3346:
3339:
3336:
3333:
3327:
3324:
3321:
3317:
3311:
3308:
3304:
3300:
3294:
3289:
3285:
3282:
3273:simplifies to
3262:
3255:
3250:
3246:
3241:
3237:
3232:
3228:
3225:
3218:
3213:
3209:
3204:
3200:
3195:
3191:
3188:
3184:
3178:
3175:
3171:
3167:
3161:
3156:
3152:
3149:
3127:
3121:
3118:
3111:
3108:
3104:
3098:
3095:
3088:
3085:
3081:
3077:
3073:
3041:
3037:
3034:
3030:
3025:
3022:
3018:
3010:
3005:
3001:
2996:
2992:
2987:
2983:
2980:
2973:
2968:
2964:
2959:
2955:
2950:
2946:
2943:
2939:
2930:
2926:
2923:
2919:
2915:
2912:
2908:
2902:
2899:
2896:
2891:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2856:
2852:
2849:
2846:
2843:
2840:
2817:
2814:
2811:
2808:
2805:
2799:
2796:
2793:
2788:
2765:
2758:
2753:
2749:
2744:
2740:
2735:
2731:
2728:
2721:
2716:
2712:
2707:
2703:
2698:
2694:
2691:
2687:
2681:
2678:
2674:
2670:
2664:
2659:
2655:
2652:
2627:
2605:
2601:
2577:
2571:
2568:
2565:
2559:
2556:
2553:
2549:
2543:
2540:
2537:
2534:
2531:
2509:
2506:
2503:
2500:
2496:
2461:
2458:
2455:
2452:
2449:
2441:
2437:
2433:
2427:
2423:
2415:
2412:
2407:
2404:
2399:
2395:
2385:simplifies to
2366:
2362:
2358:
2355:
2335:
2332:
2328:
2324:
2321:
2318:
2315:
2310:
2306:
2302:
2299:
2294:
2290:
2257:
2236:
2224:
2221:
2202:
2198:
2177:
2173:
2169:
2166:
2146:
2126:
2106:
2101:
2096:
2090:
2087:
2084:
2081:
2078:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2050:
2046:
2042:
2036:
2029:
2025:
2021:
2018:
2015:
2012:
2009:
1967:Main article:
1964:
1961:
1937:
1917:
1895:
1891:
1870:
1867:
1864:
1861:
1857:
1850:
1846:
1841:
1838:
1835:
1829:
1825:
1820:
1817:
1813:
1809:
1795:
1767:Main article:
1764:
1761:
1717:
1692:
1688:
1679:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1652:
1648:
1645:
1638:
1634:
1629:
1625:
1618:
1614:
1610:
1607:
1604:
1601:
1598:
1568:
1540:
1537:
1534:
1512:
1509:
1506:
1483:
1460:
1457:
1454:
1407:
1403:
1400:
1397:
1391:
1388:
1385:
1382:
1359:
1356:
1351:
1348:
1326:
1323:
1318:
1315:
1312:
1290:
1287:
1284:
1260:
1256:
1235:
1232:
1229:
1226:
1206:
1202:
1198:
1195:
1192:
1187:
1183:
1180:
1173:
1169:
1164:
1160:
1153:
1149:
1145:
1142:
1139:
1136:
1133:
1102:
1082:
1079:
1076:
1073:
1068:
1064:
1060:
1057:
1053:
1031:
1027:
1003:
977:
956:
936:
933:
930:
921:
917:
914:
910:
884:
861:
857:
854:
851:
848:
845:
842:
816:
812:
808:
782:
779:
727:Main article:
724:
721:
683:appears to be
622:
615:
614:
606:
599:
598:
597:
596:
595:
587:backscattering
559:
552:
551:
547:
540:
539:
538:
537:
536:
527:
524:
523:
522:
515:
508:
506:
503:
496:
494:
491:
484:
482:
479:
472:
470:
467:
460:
439:finite element
359:spherical wave
323:
320:
226:
223:
115:
91:
15:
13:
10:
9:
6:
4:
3:
2:
6521:
6510:
6507:
6505:
6502:
6501:
6499:
6489:
6479:
6477:
6467:
6465:
6460:
6455:
6453:
6443:
6441:
6436:
6431:
6429:
6424:
6419:
6415:
6407:
6404:
6400:
6396:
6392:
6388:
6386:
6383:
6382:
6378:
6370:
6366:
6361:
6356:
6352:
6348:
6344:
6340:
6336:
6329:
6326:
6321:
6317:
6313:
6309:
6305:
6301:
6297:
6293:
6289:
6285:
6281:
6277:
6273:
6266:
6263:
6258:
6254:
6248:
6244:
6243:
6235:
6232:
6227:
6221:
6217:
6210:
6207:
6202:
6196:
6192:
6185:
6183:
6179:
6175:
6169:
6164:
6163:
6154:
6151:
6148:
6147:0-444-10791-6
6144:
6140:
6134:
6132:
6128:
6123:
6119:
6115:
6111:
6110:
6102:
6099:
6094:
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6082:
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6072:
6069:
6064:
6060:
6056:
6052:
6048:
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6040:
6033:
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6025:
6019:
6015:
6014:
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6003:
5990:
5986:
5982:
5978:
5971:
5968:
5956:
5949:
5946:
5942:
5936:
5933:
5929:
5928:
5923:
5919:
5915:
5912:
5908:
5904:
5901:
5899:
5892:
5889:
5885:
5881:
5877:
5873:
5869:
5865:
5861:
5857:
5849:
5843:
5840:
5836:
5832:
5828:
5824:
5818:
5815:
5809:
5805:
5800:
5795:
5791:
5787:
5783:
5779:
5778:
5773:
5766:
5763:
5759:
5755:
5752:
5748:
5744:
5738:
5735:
5730:
5726:
5725:
5717:
5714:
5710:
5706:
5702:
5701:
5693:
5690:
5686:
5682:
5678:
5674:
5671:
5667:
5661:
5658:
5647:
5641:
5637:
5633:
5629:
5625:
5618:
5615:
5610:
5606:
5602:
5598:
5594:
5590:
5586:
5582:
5578:
5574:
5569:
5564:
5560:
5556:
5548:
5545:
5540:
5538:9780122274107
5534:
5530:
5526:
5522:
5515:
5512:
5500:
5494:
5490:
5489:
5481:
5478:
5472:
5469:
5465:
5461:
5458:
5452:
5449:
5445:
5443:
5437:
5435:
5429:
5425:
5422:
5418:
5412:
5409:
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5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5279:
5274:
5272:
5270:
5262:
5257:
5255:
5252:
5248:
5246:
5242:
5237:
5232:
5224:
5222:
5220:
5215:
5201:
5193:
5189:
5185:
5180:
5178:
5162:
5142:
5122:
5102:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5050:
5046:
5042:
5034:
5030:
5025:
5020:
5012:
5010:
5007:
5002:
5000:
4984:
4976:
4960:
4940:
4934:
4931:
4926:
4923:
4916:
4909:
4905:
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4893:
4888:
4884:
4881:
4878:
4873:
4872:
4871:
4864:
4857:
4848:
4843:
4836:
4831:
4829:
4827:
4823:
4819:
4815:
4810:
4806:
4801:
4799:
4795:
4791:
4787:
4783:
4776:"Knife edge"
4775:
4773:
4770:
4765:
4755:
4752:
4751:laser pointer
4748:
4742:
4734:
4732:
4730:
4726:
4721:
4718:
4712:
4710:
4694:
4674:
4669:
4666:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4635:
4631:
4615:
4612:
4609:
4589:
4569:
4561:
4545:
4525:
4505:
4502:
4499:
4496:
4493:
4490:
4479:
4471:
4468:
4464:
4463:
4457:
4452:
4444:
4442:
4438:
4436:
4435:self-focusing
4432:
4428:
4427:Gaussian beam
4424:
4420:
4412:
4410:
4408:
4403:
4399:
4394:
4381:
4377:
4374:
4370:
4365:
4362:
4358:
4348:
4345:
4339:
4335:
4331:
4327:
4324:
4318:
4314:
4307:
4304:
4300:
4292:
4289:
4285:
4281:
4278:
4257:
4222:
4215:
4212:
4209:
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4187:
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4168:
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4057:
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4028:
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3991:
3988:
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3910:
3875:
3868:
3865:
3862:
3856:
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3846:
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3808:
3793:
3790:
3787:
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3760:
3754:
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3730:
3727:
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3667:
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3189:
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2672:
2650:
2642:
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2333:
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2304:
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2297:
2292:
2279:
2275:
2274:wave equation
2272:
2234:
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2218:
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2196:
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2171:
2167:
2164:
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1970:
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1427:
1425:
1422:which is the
1405:
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416:
415:wave equation
412:
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356:
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319:
317:
313:
309:
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301:
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293:
289:
285:
281:
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273:
272:James Gregory
269:
265:
261:
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254:
250:
246:
242:
236:
231:
224:
222:
220:
219:semi-circular
214:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
167:
165:
161:
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6101:
6087:(3): 29–40.
6084:
6080:
6071:
6046:
6042:
6032:
6012:
6005:
5993:. Retrieved
5980:
5970:
5960:21 September
5958:. Retrieved
5948:
5940:
5935:
5925:
5921:
5897:
5891:
5875:
5871:
5859:
5850:, vol.
5847:
5842:
5826:
5822:
5817:
5781:
5775:
5765:
5742:
5737:
5723:
5716:
5708:
5699:
5692:
5684:
5680:
5665:
5660:
5649:, retrieved
5627:
5617:
5558:
5554:
5547:
5520:
5514:
5502:. Retrieved
5487:
5480:
5471:
5451:
5441:
5440:
5433:
5432:
5416:
5411:
5357:Fresnel zone
5266:
5258:Applications
5253:
5249:
5238:
5234:
5216:
5181:
5176:
5048:
5038:
5032:
5006:matter waves
5003:
4911:
4869:
4802:
4785:
4781:
4779:
4767:
4744:
4722:
4713:
4475:
4460:
4439:
4416:
4395:
4058:
3619:
3402:
3139:
3061:
2777:
2589:
2473:
2247:at location
2226:
2157:is equal to
1991:
1978:
1957:
1950:
1796:
1792:
1789:
1705:
1557:
1439:
1428:
1124:equation as
1115:
1016:
796:
764:
756:
718:
711:
708:
693:
674:
648:
633:
573:
566:
529:
451:
447:
408:
396:G. I. Taylor
383:wavefunction
372:
355:point source
336:
290:performed a
288:Thomas Young
267:
264:Isaac Newton
255:
248:
239:
215:
193:, and other
177:, or when a
168:
156:interference
129:
68:
48:
47:
25:
6504:Diffraction
6488:Outer space
6476:Spaceflight
6440:Mathematics
5995:9 September
5442:Translation
5372:Quasioptics
5049:Bragg's law
5029:Bragg's law
4900:Matter wave
4887:double-slit
4478:diffraction
4470:zeta Boötis
4467:binary star
4462:lucky image
4431:convex lens
3620:Now, since
1953:convolution
1587:path length
454:water waves
332:ripple tank
256:diffringere
251:, from the
249:diffraction
203:radio waves
191:water waves
69:diffraction
61:propagating
49:Diffraction
6498:Categories
6408:at YouTube
5404:References
5382:Reflection
5377:Refraction
5033:reflection
5027:Following
4898:See also:
4805:half-plane
4419:laser beam
1444:above, if
685:iridescent
643:Arago spot
591:refraction
518:spider web
431:near field
367:amplitudes
179:sound wave
171:light wave
152:wavelength
21:refraction
6452:Astronomy
6304:1476-4687
5808:110408369
5593:1748-3395
5568:1402.1867
5523:: 67–85.
5504:7 January
5225:Coherence
5192:electrons
5143:θ
5103:λ
5080:θ
5077:
5062:λ
4924:λ
4798:wavefront
4790:radiation
4667:λ
4656:θ
4653:
4647:≈
4644:θ
4613:≫
4526:λ
4500:λ
4488:Δ
4305:−
4223:∬
4213:π
4188:∝
4176:Ψ
4149:ϕ
4146:
4140:θ
4137:
4098:ϕ
4095:
4089:θ
4086:
4005:ϕ
4002:
3985:ϕ
3982:
3973:θ
3970:
3958:−
3876:∬
3866:π
3841:∝
3829:Ψ
3803:^
3794:θ
3791:
3779:^
3767:ϕ
3764:
3755:θ
3752:
3740:^
3731:ϕ
3728:
3722:θ
3719:
3707:^
3678:^
3655:^
3571:^
3562:⋅
3540:−
3458:∬
3448:π
3423:∝
3411:Ψ
3380:^
3371:⋅
3349:−
3335:π
3281:ψ
3240:−
3227:π
3203:−
3148:ψ
3120:^
3097:^
2995:−
2982:π
2958:−
2855:∬
2851:∝
2839:Ψ
2743:−
2730:π
2706:−
2651:ψ
2567:π
2530:ψ
2505:ω
2499:−
2454:ψ
2432:∂
2422:∂
2403:ψ
2394:∇
2354:δ
2320:δ
2314:ψ
2298:ψ
2289:∇
2235:ψ
2176:λ
2168:π
2089:θ
2086:
2069:θ
2066:
2014:θ
1998:variation
1994:Airy disk
1979:Airy disk
1969:Airy disk
1890:θ
1866:λ
1845:θ
1840:
1834:±
1824:θ
1819:
1716:θ
1678:θ
1674:
1668:±
1665:θ
1662:
1651:λ
1647:π
1633:
1603:θ
1567:θ
1536:≈
1533:θ
1511:λ
1508:≫
1482:θ
1459:λ
1456:≪
1440:From the
1431:far field
1399:
1384:
1355:π
1347:θ
1322:π
1317:−
1311:θ
1283:θ
1231:θ
1197:θ
1194:
1186:λ
1182:π
1168:
1138:θ
1078:λ
1063:θ
1059:
1044:given by
1026:θ
1002:λ
976:θ
932:λ
920:θ
916:
901:given by
883:θ
853:θ
847:
807:λ
781:π
681:deli meat
445:methods.
423:far field
322:Mechanism
268:inflexion
162:(e.g., a
140:wavefront
114:θ
104:wavefront
36:of a red
6369:24914146
6312:10963603
6257:Archived
5989:Archived
5914:Archived
5903:Archived
5784:: 1–16.
5754:Archived
5673:Archived
5670:pp. 1–11
5651:25 April
5601:22447163
5460:Archived
5434:Original
5424:Archived
5275:See also
5188:neutrons
4999:momentum
4858:Patterns
4630:paraxial
4560:f-number
4437:effect.
4423:coherent
4378:′
4366:′
4349:′
4328:′
4293:′
4282:′
4059:Letting
4041:′
4029:′
4012:′
3992:′
3946:′
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3645:′
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3602:′
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3024:′
3004:′
2967:′
2925:′
2914:′
2752:′
2715:′
2673:′
2604:′
768:coherent
654:aperture
623:A solar
569:hologram
526:Examples
349:and the
241:Da Vinci
197:such as
148:coherent
144:wavelets
53:aperture
42:aperture
6428:Physics
6414:Portals
6360:4052855
6320:4300920
6284:Bibcode
6051:Bibcode
5786:Bibcode
5609:5918772
5573:Bibcode
4997:is the
4973:is the
4828:(UTD).
4558:is the
4465:of the
1996:. The
696:jetties
425:), the
417:), the
373:In the
225:History
6367:
6357:
6318:
6310:
6302:
6276:Nature
6249:
6222:
6216:Optics
6197:
6170:
6145:
6020:
5927:Traité
5884:337–78
5880:246–96
5806:
5751:p. 254
5642:
5607:
5599:
5591:
5535:
5495:
5421:page 2
5184:X-rays
5095:where
4953:where
4906:, and
4687:where
4518:where
3770:
3758:
2932:
2346:where
2117:where
1881:where
1217:where
1093:where
947:where
609:corona
607:Lunar
579:corona
379:photon
363:phases
199:X-rays
57:shadow
6464:Stars
6316:S2CID
5981:Slate
5911:p. 15
5804:S2CID
5605:S2CID
5563:arXiv
5190:(and
5177:order
4877:sines
2522:) is
2474:(See
2215:is a
1525:only
625:glory
583:glory
449:out.
343:waves
253:Latin
38:laser
6365:PMID
6308:PMID
6300:ISSN
6247:ISBN
6220:ISBN
6195:ISBN
6168:ISBN
6143:ISBN
6018:ISBN
5997:2013
5962:2007
5653:2024
5640:ISBN
5597:PMID
5589:ISSN
5533:ISBN
5506:2023
5493:ISBN
4977:and
4780:The
4745:The
4662:1.22
4497:1.22
4110:and
3693:and
2188:and
1624:sinc
1381:sinc
1159:sinc
689:wave
675:The
441:and
400:1909
308:1818
306:and
304:1816
296:1803
280:1675
276:1638
260:1665
235:1803
201:and
73:1660
6355:PMC
6347:doi
6343:369
6292:doi
6280:406
6118:doi
6089:doi
6059:doi
5920::
5794:doi
5705:149
5679::
5632:doi
5581:doi
5525:doi
5430::
5074:sin
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4650:sin
4636:is
4409:).
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4134:sin
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4083:sin
3999:sin
3979:cos
3967:sin
3788:cos
3761:sin
3749:sin
3725:cos
3716:sin
2280:),
2083:sin
2063:sin
1837:sin
1816:sin
1671:sin
1659:sin
1396:sin
1339:to
1191:sin
1056:sin
980:min
924:min
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887:min
844:sin
398:in
337:In
294:in
130:In
6500::
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5810:.
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4927:=
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4675:,
4670:D
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4628:(
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4506:,
4503:N
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4491:x
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2301:+
2293:2
2256:r
2201:1
2197:J
2172:/
2165:2
2145:k
2125:a
2105:,
2100:2
2095:)
2080:a
2077:k
2072:)
2060:a
2057:k
2054:(
2049:1
2045:J
2041:2
2035:(
2028:0
2024:I
2020:=
2017:)
2011:(
2008:I
1936:m
1916:d
1894:i
1869:,
1863:m
1860:=
1856:)
1849:i
1828:m
1812:(
1808:d
1797:m
1793:θ
1734:.
1720:i
1691:]
1687:)
1682:i
1656:(
1644:d
1637:[
1628:2
1617:0
1613:I
1609:=
1606:)
1600:(
1597:I
1571:i
1539:0
1523:,
1505:d
1494:,
1471:,
1453:d
1433:(
1420:,
1406:x
1402:x
1390:=
1387:x
1358:2
1350:=
1325:2
1314:=
1289:0
1286:=
1275:(
1259:0
1255:I
1234:)
1228:(
1225:I
1205:,
1201:)
1179:d
1172:(
1163:2
1152:0
1148:I
1144:=
1141:)
1135:(
1132:I
1101:n
1081:,
1075:n
1072:=
1067:n
1052:d
1030:n
955:d
935:,
929:=
909:d
860:2
856:)
850:(
841:d
827:.
815:2
811:/
778:2
611:.
278:–
274:(
90:d
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