3081:[When an opaque body is placed in a beam of light, its shadow is bordered on the outside by bands of various shades and widths. These bands were studied by Newton in the first book of his Optics; but this famous physicist does not speak of the no less remarkable bands which form in the interior of the shadow of loose bodies, although Grimaldi had already given a detailed description of them in his work, and he even affirms positively that no light enters the geometric shadow. The inaccuracy of this result was sufficiently proven by Maraldi and De l'Isle, who, moreover, added nothing salient to what Grimaldi had discovered long before.]
2569:
2564:
2559:
2145:
2140:
2135:
77:
1957:
1952:
1947:
423:
85:
1020:
101:
3062:(One of your commissioners, Mr. Poisson, had deduced from the integrals reported by the author , the strange result that the center of the shadow of an opaque circular screen should — when the rays penetrate it at slightly oblique incidences — also be illuminated as if the screen didn't exist. This result has been submitted to the test of a direct experiment, and observation has perfectly confirmed the calculation (e).)
3060:"L'un de vos commissaires, M. Poisson, avait déduit des intégrales rapportées par l'auteur, le résultat singulier que le centre de l'ombre d'un écran circulaire opaque devait, lorsque les rayons y pénétraient sous des incidences peu obliques, être aussi éclairé que si l'écran n'existait pas. Cette conséquence a été soumise à l'épreuve d'une expérience directe, et l'observation a parfaitement confirmé le calcul (e)."
414:. Today it is known that a diffraction pattern appears through the mosaic-like buildup of bright spots caused by single photons, as predicted by Dirac's quantum theory. With increasing light intensity the bright dots in the mosaic diffraction pattern just assemble faster. In contrast, the wave theory predicts the formation of an extended continuous pattern whose overall brightness increases with light intensity.
25:
3079:
qui se forment dans l'intérieur de l'ombre des corps déliés, quoique
Grimaldi en eût déjà donné une description détaillée dans son ouvrage, et il affirme même positivement qu'aucune lumière ne pénètre dans l'ombre géométrique. L'inexactitude de ce résultat fut suffisamment prouvée par Maraldi et De l'Isle, qui, du reste, n'ajoutèrent rien de saillant à ce que Grimaldi avait découvert longtemps avant.
109:
121:
2582:. The field transmitted by a radial segment that stems from a point on the obstacle edge provides a contribution whose phase is tight to the position of the edge point relative to Fresnel zones. If the variance in the radius of the obstacle are much smaller than the width of Fresnel zone near the edge, the contributions form radial segments are approximately in phase and
1015:
2813:"Although this phenomenon is often called Poisson's spot, Poisson probably was not happy to have seen it because it supported the wave model of light. The spot is sometimes called Fresnel's spot because it is a direct consequence of his work, and Arago's spot because Arago devised the experiment that confirmed its existence." Katz, Debora M.,
1665:
3078:
Lorsqu'un corps opaque est placé dans un faisceau de lumière, son ombre est bordée à l'extérieur de bandes de diverses nuances et de diverses largeurs. Ces bandes ont été étudiées par Newton dans le premier livre de son
Optique; mais ce célèbre physicien ne parle pas des bandes non moins remarquables
2542:
The Arago spot is very sensitive to small-scale deviations from the ideal circular cross-section. This means that a small amount of surface roughness of the circular object can completely cancel out the bright spot. This is shown in the following three diagrams which are simulations of the Arago spot
1962:
The images are simulations of the Arago spot in the shadow of discs of diameter 4 mm, 2 mm, and 1 mm, imaged 1 m behind each disc. The disks are illuminated by light of wavelength of 633 nm, diverging from a point 1 m in front of each disc. Each image is 16 mm wide.
1453:
To calculate the full diffraction image that is visible on the screen one has to consider the surface integral of the previous section. One cannot exploit circular symmetry anymore, since the line between the source and an arbitrary point on the screen does not pass through the center of the circular
3126:(More light in the middle of the smaller balls shows that it spreads in greater abundance and more easily around small balls than around big .) Fig. 8 on Plate 6 (following p. 142) shows light at the center of a ball's shadow (in the regime where the spot and its fringes cover most of the shadow).
361:
Poisson studied
Fresnel's theory in detail and, being a supporter of the particle theory of light, looked for a way to prove it wrong. Poisson thought that he had found a flaw when he argued that a consequence of Fresnel's theory was that there would exist an on-axis bright spot in the shadow of a
817:
406:, has nothing in common with the particles imagined in the corpuscular theory that had been dominant before the rise of the wave theory and Arago's powerful demonstration. Before the advent of quantum theory in the late 1920s, only the wave nature of light could explain phenomena such as
2525:
If the cross-section of the circular object deviates slightly from its circular shape (but it still has a sharp edge on a smaller scale) the shape of the point-source Arago spot changes. In particular, if the object has an ellipsoidal cross-section the Arago spot has the shape of an
1493:
127:
126:
123:
122:
2330:
2714:
Beside the demonstration of wave-behavior, the Arago spot also has a few other applications. One of the ideas is to use the Arago spot as a straight line reference in alignment systems. Another is to probe aberrations in laser beams by using the spot's sensitivity to beam
128:
1203:
1681:. However, it is possible to solve the radial part of the integral so that only the integration over the azimuth angle remains to be done numerically. For a particular angle one must solve the line integral for the ray with origin at the intersection point of the line P
602:
1984:. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. This means that one can compensate for a reduction in the source's
362:
circular obstacle, where there should be complete darkness according to the particle theory of light. This prediction was seen as an absurd consequence of the wave theory, and the failure of that prediction should be a strong argument to reject
Fresnel's theory.
1875:
2586:
constructively. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent
Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity.
125:
2574:
The simulation includes a regular sinusoidal corrugation of the circular shape of amplitude 10 ÎĽm, 50 ÎĽm and 100 ÎĽm, respectively. Note, that the 100 ÎĽm edge corrugation almost completely removes the central bright spot.
1489:
which is 1 for transparent parts of the object plane and 0 otherwise (i.e. It is 0 if the direct line between source and the point on the screen passes through the blocking circular object.) the integral that needs to be solved is given by:
369:, decided to actually perform the experiment. He molded a 2 mm metallic disk to a glass plate with wax. He succeeded in observing the predicted spot, which convinced most scientists of the wave nature of light and gave Fresnel the win.
2092:
2670:
The edge corrugation should not be much more than 10% of this width to see a close to ideal Arago spot. In the above simulations with the 4 mm diameter disc the adjacent
Fresnel zone has a width of about 77 ÎĽm.
2666:
2201:
2516:
707:
1062:
1287:
453:
2158:
The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. If the wave source has a finite size
1010:{\displaystyle U(P_{1})=-{\frac {\mathbf {i} }{\lambda }}{\frac {Ae^{\mathbf {i} k(g+b)}}{gb}}2\pi \int _{a}^{\infty }e^{\mathbf {i} k{\frac {1}{2}}\left({\frac {1}{g}}+{\frac {1}{b}}\right)r^{2}}r\,dr.}
1771:
235:
2171:, as if the circular object acted like a lens. At the same time the intensity of the Arago spot is reduced with respect to the intensity of the undisturbed wave front. Defining the relative intensity
3046:[Report made by Mr. Arago to the Academy of Sciences in the name of the commission which had been charged with examining the memoirs submitted to the competition for the diffraction prize.].
2150:
The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above.
2198:
as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation:
2530:. Note that this is only the case if the source is close to an ideal point source. From an extended source the Arago spot is only affected marginally, since one can interpret the Arago spot as a
1660:{\displaystyle U(P_{1})\propto \int _{0}^{2\pi }\int _{0}^{\infty }g(r,\theta )e^{{\frac {\mathbf {i} \pi \rho ^{2}}{\lambda }}\left({\frac {1}{g}}+{\frac {1}{b}}\right)}\rho \,d\rho \,d\theta .}
1425:
333:
in 1807. The original Arago spot experiment was carried out a decade later and was the deciding experiment on the question of whether light is a particle or a wave. It is thus an example of an
1346:
754:
3124:"La lumiere plus grande au milieu des boules plus petites, fait voir qu'elle circule en plus grande abondance & plus facilement autour des petites boules qu'autour des grandes."
124:
3102:
Delisle mentions that when a small ball was illuminated by sunlight, the ball's shadow contained alternating bright and dark rings concentric with the center of the ball's shadow.
3044:"Rapport fait par M. Arago à l'Académie des Sciences, au nom de la Commission qui avait été chargée d'examiner les Mémoires envoyés au concours pour le prix de la diffraction"
2196:
1917:
2534:. Therefore, the image of the extended source only becomes washed out due to the convolution with the point-spread function, but it does not decrease in overall intensity.
1487:
1433:
much greater than the diameter of the circular obstacle is the same as the source intensity, as if the circular object was not present at all. However at larger distances
1714:
2006:
446:
at a point on the screen is given by the superposition of all those secondary wavelets taking into account their relative phases. This means that the field at a point P
2424:
631:
2384:
2357:
1940:
1053:
3094:"Sur l'expérience que j'ai rapportée à l'Academie d'un anneau lumineux semblable à celui que l'on apperçoit autour de la lune dans les eclipses totales du soleil"
2593:
1766:
1740:
2444:
2404:
3169:
2449:
786:
In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the
636:
301:
of the obstacle and going to the center of the shadow travels exactly the same distance, so all the light passing close by the object arrives at the screen in
3096:[On the experience that I reported to the Academy about a luminous ring similar to that which one sees around the moon during a total solar eclipse].
442:, which states that every unobstructed point of a wavefront becomes the source of a secondary spherical wavelet and that the amplitude of the optical field
3563:
88:
Arago spot experiment. A point source illuminates a circular object, casting a shadow on a screen. At the shadow's center a bright spot appears due to
186:
383:
Although Arago's experimental result was overwhelming evidence in favor of the wave theory, a century later, in conjunction with the birth of
350:
launched a competition to explain the properties of light, where
Poisson was one of the members of the judging committee. The civil engineer
1355:
1214:
112:
Numerical simulation of the intensity of monochromatic light of wavelength λ = 0.5 μm behind a circular obstacle of radius
3538:
3386:
Reisinger, Thomas; Patel, A. Amil; Reingruber, Herbert; Fladischer, Katrin; Ernst, Wolfgang E.; Bracco, Gianangelo; Smith, Henry I.;
2325:{\displaystyle I_{\text{rel}}(w)=J_{0}^{2}\left({\frac {wR\pi }{g\lambda }}\right)+J_{1}^{2}\left({\frac {wR\pi }{g\lambda }}\right)}
1198:{\displaystyle U(P_{1})={\frac {Ae^{\mathbf {i} kg}}{g}}{\frac {b}{\sqrt {b^{2}+a^{2}}}}e^{\mathbf {i} k{\sqrt {b^{2}+a^{2}}}}.}
398:), it became understood that light (as well as all forms of matter and energy) must be described as both a particle and a wave (
3391:
597:{\displaystyle U(P_{1})={\frac {Ae^{\mathbf {i} kr_{0}}}{r_{0}}}\int _{S}{\frac {e^{\mathbf {i} kr_{1}}}{r_{1}}}K(\chi )\,dS,}
2875:
2847:
330:
372:
Arago later noted that the phenomenon (later known as "Poisson's spot" or the "spot of Arago") had already been observed by
297:
says that every point in the plane of the obstacle acts as a new point source of light. The light coming from points on the
325:
At the beginning of the 19th century, the idea that light does not simply propagate along straight lines gained traction.
2770:
2723:
has been proposed as a method for dramatically improving the diffraction-limited resolution of space-based telescopes.
2706:
The observation of an Arago spot with large molecules, thus proving their wave-nature, is a topic of current research.
1023:
The on-axis intensity at the center of the shadow of a small circular obstacle converges to the unobstructed intensity.
3209:
Dauger, D.E. (November 1996), "Simulation and Study of
Fresnel Diffraction for Arbitrary Two-Dimensional Apertures",
3182:
3152:
2975:
2935:
2911:
2799:
1870:{\displaystyle R(\theta _{1})\propto e^{{\frac {\pi }{2}}\mathbf {i} s^{2}}-e^{{\frac {\pi }{2}}\mathbf {i} t^{2}}.}
2700:
3343:
Coulson, John; Becknell, G. G. (1922), "Reciprocal
Diffraction Relations between Circular and Elliptical Plates",
790:
regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances
2583:
1292:
306:
56:
2822:
2696:
776:. Similarly, the term inside the integral represents the oscillations from the secondary wavelets at distances
3249:
1988:
by increasing the distance between the circular object and screen or reducing the circular object's diameter.
721:
439:
263:
These conditions together explain why the bright spot is not encountered in everyday life. However, with the
3056:
3043:
3072:
Arago, F. "Mémoire sur la méthode des interférences appliquée à la recherche des indices de réfraction.".
2446:
the distance between source and disc. For large sources the following asymptotic approximation applies:
3174:
347:
52:
2130:
The following images show the radial intensity distribution of the simulated Arago spot images above:
769:
The first term outside of the integral represents the oscillations from the source wave at a distance
3352:
399:
2174:
1886:
171:
The basic experimental setup requires a point source, such as an illuminated pinhole or a diverging
3113:
342:
At that time, many favored Isaac Newton's corpuscular theory of light, among them the theoretician
326:
314:
3118:
Histoire de l'Académie Royale des
Sciences ... Avec les Mémoires de Mathématique & de Physique
3098:
Histoire de l'Académie Royale des Sciences ... Avec les Mémoires de Mathématique & de Physique
1457:
2950:
2903:
2892:
2762:
1692:
16:
Bright point that appears at the center of a circular object's shadow due to Fresnel diffraction.
3093:
343:
392:
373:
3030:
2409:
607:
3007:
2815:
Physics for Scientists and Engineers: Foundations and Connections, Advance Edition, Volume 2,
2568:
2531:
351:
294:
3526:
2563:
2558:
3595:
3494:
3450:
3406:
3356:
3307:
3261:
3218:
2867:
2362:
2335:
1922:
1677:
is not efficient and becomes numerically unstable especially for configurations with large
1038:
355:
1445:
is increased in successive images) is larger therefore making the spot easier to discern.
8:
2993:
Opticks: Or, A Treatise of the Reflections, Refractions, Inflections and Colours of Light
1745:
1719:
1689:
with the circular object plane. The contribution for a particular ray with azimuth angle
1208:
787:
335:
279:
176:
157:
3498:
3454:
3410:
3360:
3311:
3265:
3222:
2679:
In 2009, the Arago spot experiment was demonstrated with a supersonic expansion beam of
3510:
3468:
3141:
2716:
2429:
2389:
377:
310:
3564:"New space telescope concept could image objects at far higher resolution than Hubble"
2144:
2139:
2134:
3325:
3178:
3148:
2971:
2931:
2907:
2871:
2843:
2839:
2818:
2795:
2766:
2699:. An Arago spot of electrons, which also constitute matter waves, can be observed in
2688:
1674:
807:
411:
384:
37:
3485:
Davisson, C.; Germer, L. (1927), "Diffraction of Electrons by a Crystal of Nickel",
1991:
The lateral intensity distribution on the screen has in fact the shape of a squared
366:
3514:
3502:
3472:
3458:
3422:
3414:
3364:
3315:
3269:
3226:
2692:
2087:{\displaystyle U(P_{1},r)\propto J_{0}^{2}\left({\frac {\pi rd}{\lambda b}}\right)}
1670:
287:
93:
3011:
2991:
2965:
1992:
633:
which ensures that the secondary wavelets do not propagate backwards is given by
388:
76:
3418:
3320:
3295:
2791:
2737:
1678:
180:
84:
1956:
1951:
1946:
3589:
3329:
302:
298:
2661:{\displaystyle \Delta r\approx {\sqrt {r^{2}+\lambda {\frac {gb}{g+b}}}}-r.}
274:, the Arago spot can also be observed in the strongly defocussed image of a
3368:
2684:
2579:
1996:
1977:
283:
3387:
3277:
2899:
2511:{\displaystyle I_{\text{rel}}(w)\approx {\frac {2g\lambda }{\pi ^{2}wR}}}
702:{\displaystyle K(\chi )={\frac {\mathbf {i} }{2\lambda }}(1+\cos(\chi ))}
422:
407:
161:
89:
45:
3427:
267:
available today, it is undemanding to perform an Arago-spot experiment.
2732:
2720:
2000:
1985:
1981:
1019:
757:
172:
80:
Photo of the Arago spot in a shadow of a 5.8 mm circular obstacle.
41:
260:
Finally, the edge of the circular object must be sufficiently smooth.
100:
3506:
3463:
3231:
2680:
1883:) of the ray with the circular object and then sum the contributions
1282:{\textstyle I_{0}=\left|{\frac {1}{g}}Ae^{\mathbf {i} kg}\right|^{2}}
271:
3273:
1055:
is not negligible one can write the integral for the on-axis case (P
175:. The dimensions of the setup must comply with the requirements for
152:
is a bright point that appears at the center of a circular object's
132:
Formation of the Arago spot (select "WebM source" for good quality).
1942:. Results of such a calculation are shown in the following images.
402:). However the particle associated with electromagnetic waves, the
3385:
2967:
A Course of Lectures on Natural Philosophy and the Mechanical Arts
1441:
of the bright spot (as can be seen in the simulations below where
2834:
Pedrotti, Frank L.; Pedrotti, Leno S.; Pedrotti, Leno M. (2007),
2527:
1980:, the intensity of the Arago spot equals that of the undisturbed
168:
and is a common way to demonstrate that light behaves as a wave.
3147:(in German), vol. 4 (3rd ed.), Verlag Harri Deutsch,
403:
153:
137:
309:. This results in a bright spot at the shadow's center, where
3294:
Reisinger, T; Leufke, P M; Gleiter, H; Hahn, H (2017-03-14).
2786:
Hecht, Eugene; Zajac, Alfred (1974), "10.3, "Diffraction,"",
264:
165:
160:. This spot played an important role in the discovery of the
108:
1879:
So for each angle one has to compute the intersection point(
2153:
275:
3539:"The Aragoscope: Ultra-High Resolution Optics at Low Cost"
3293:
230:{\displaystyle F={\frac {d^{2}}{\ell \lambda }}\gtrsim 1,}
2757:
Law, Jonathan; Rennie, Richard (2015), "Poisson's Spot",
2163:
then the Arago spot will have an extent that is given by
810:
then yields the integral for a circular object of radius
3250:"The spot of Arago: New relevance for an old phenomenon"
2833:
2687:). Material particles behaving like waves is known from
1716:
and passing a transparent part of the object plane from
1027:
This integral can be solved numerically (see below). If
2537:
2703:
when examining circular structures of a certain size.
2691:. The wave nature of particles actually dates back to
1348:. The on-axis intensity as a function of the distance
1217:
724:
426:
Notation for calculating the wave amplitude at point P
250:
is the distance between the object and the screen, and
2596:
2590:
The adjacent Fresnel zone is approximately given by:
2452:
2432:
2412:
2392:
2365:
2338:
2204:
2177:
2009:
1925:
1889:
1774:
1748:
1722:
1695:
1496:
1460:
1358:
1295:
1065:
1059:
is at the center of the shadow) as (see Sommerfeld):
1041:
820:
639:
610:
456:
189:
1448:
1420:{\displaystyle I={\frac {b^{2}}{b^{2}+a^{2}}}I_{0}.}
290:
of the telescope constitutes the circular obstacle.
2126:
is the distance between circular object and screen.
1429:This shows that the on-axis intensity at distances
3140:
2891:
2660:
2510:
2438:
2418:
2398:
2378:
2351:
2324:
2190:
2086:
1934:
1911:
1869:
1760:
1734:
1708:
1659:
1481:
1419:
1340:
1281:
1197:
1047:
1009:
748:
701:
625:
596:
229:
3020:
1211:, which is the square of the field amplitude, is
3587:
3381:
3379:
3377:
2930:(4th ed.), Pearson Education, p. 494,
2838:(3rd ed.), Upper Saddle River, New Jersey:
1669:Numerical calculation of the integral using the
3342:
2674:
3484:
3441:de Broglie, Louis (1923), "Waves and Quanta",
3190:
2779:
2406:is the radius of the disc casting the shadow,
450:on the screen is given by a surface integral:
317:predict that there should be no light at all.
3374:
3296:"On the relative intensity of Poisson's spot"
3247:
3243:
3241:
3134:
3132:
2578:This effect can be best understood using the
2520:
1919:for a certain number of angles between 0 and
438:At the heart of Fresnel's wave theory is the
354:entered this competition by submitting a new
3248:Harvey, James E.; Forgham, James L. (1984),
2386:are the Bessel functions of the first kind.
293:When light shines on the circular obstacle,
3336:
3143:Vorlesungen ĂĽber Theoretische Physik: Optik
3105:
3085:
3000:
2827:
1341:{\displaystyle I=\left|U(P_{1})\right|^{2}}
282:. There, the star provides an almost ideal
3440:
3238:
3196:
3138:
3129:
2983:
2883:
2785:
3462:
3434:
3426:
3319:
3230:
2855:
2756:
1647:
1640:
997:
584:
3478:
3405:(5), American Physical Society: 053823,
3166:
3160:
2957:
2919:
2154:Finite source size and spatial coherence
1993:zeroth Bessel function of the first kind
1018:
749:{\textstyle k={\frac {2\pi }{\lambda }}}
421:
119:
107:
99:
83:
75:
3116:[Various optical experiments].
3111:
3091:
3026:
3006:
2889:
1966:
244:is the diameter of the circular object,
3588:
3208:
3202:
2989:
2861:
3289:
3287:
3041:
2963:
2925:
2709:
1971:
2538:Surface roughness of circular object
365:However, the head of the committee,
18:
2108:on the screen from the optical axis
1454:object. With the aperture function
716:is the amplitude of the source wave
13:
3284:
2597:
2114:is the diameter of circular object
1547:
921:
430:from a spherical point source at P
92:, contradicting the prediction of
23:
14:
3607:
3071:
2701:transmission electron microscopes
2697:Davisson and Germer's experiments
2683:molecules (an example of neutral
1449:Calculation of diffraction images
814:(see for example Born and Wolf):
104:Arago spot forming in the shadow.
3114:"Diverses expèriences d'optique"
3048:Annales de Chimie et de Physique
2567:
2562:
2557:
2543:from a 4 mm diameter disc (
2143:
2138:
2133:
1955:
1950:
1945:
1848:
1813:
1579:
1289:and the intensity at the screen
1256:
1156:
1100:
932:
870:
849:
658:
541:
491:
256:is the wavelength of the source.
3556:
3531:
3520:
3392:"Poisson's spot with molecules"
3065:
3035:
387:(and first suggested in one of
3167:Born, Max; Wolf, Emil (1999),
2943:
2807:
2750:
2469:
2463:
2221:
2215:
2191:{\displaystyle I_{\text{rel}}}
2032:
2013:
1912:{\displaystyle I(\theta _{1})}
1906:
1893:
1791:
1778:
1567:
1555:
1513:
1500:
1476:
1464:
1324:
1311:
1082:
1069:
889:
877:
837:
824:
696:
693:
687:
672:
649:
643:
620:
614:
581:
575:
473:
460:
1:
3016:, Paris: Imprimerie impériale
2743:
2101:is the distance of the point
604:where the inclination factor
367:Dominique-François-Jean Arago
2675:Arago spot with matter waves
2003:(point source at infinity):
1482:{\displaystyle g(r,\theta )}
765:is the unobstructed surface.
7:
3254:American Journal of Physics
3139:Sommerfeld, Arnold (1978),
2726:
1709:{\displaystyle \theta _{1}}
1035:is small so that the angle
10:
3614:
3419:10.1103/PhysRevA.79.053823
3175:Cambridge University Press
3173:(7th, expanded ed.),
3050:. 2nd series (in French).
2970:, London: Joseph Johnson,
2521:Deviation from circularity
348:French Academy of Sciences
320:
315:particle theories of light
57:the image placement policy
36:may require adjustment of
3353:American Physical Society
440:Huygens–Fresnel principle
417:
3321:10.1088/1367-2630/aa5e7f
2817:Cengage Learning, 2015.
2419:{\displaystyle \lambda }
1437:, it turns out that the
626:{\displaystyle K(\chi )}
59:for further information.
3092:Delisle, J.-N. (1715).
2996:, London: Royal Society
2864:Fundamentals of Physics
2763:Oxford University Press
2759:A Dictionary of Physics
3369:10.1103/PhysRev.20.594
3300:New Journal of Physics
3112:Maraldi, G.F. (1723).
2990:Newton, Isaac (1704),
2964:Young, Thomas (1807),
2926:Hecht, Eugene (2002),
2890:Ohanian, Hans (1989),
2862:Walker, Jearl (2008),
2836:Introduction to Optics
2695:hypothesis as well as
2662:
2512:
2440:
2420:
2400:
2380:
2353:
2326:
2192:
2088:
1936:
1913:
1871:
1762:
1736:
1710:
1661:
1483:
1421:
1342:
1283:
1199:
1049:
1024:
1011:
788:near-field diffraction
750:
703:
627:
598:
435:
331:double-slit experiment
231:
133:
117:
105:
97:
81:
34:This article's images
28:
3120:(in French): 111–143.
3100:(in French): 166–169.
2868:John Wiley & Sons
2663:
2532:point-spread function
2513:
2441:
2421:
2401:
2381:
2379:{\displaystyle J_{1}}
2354:
2352:{\displaystyle J_{0}}
2327:
2193:
2089:
1937:
1935:{\displaystyle 2\pi }
1914:
1872:
1763:
1737:
1711:
1662:
1484:
1422:
1343:
1284:
1200:
1050:
1048:{\displaystyle \chi }
1022:
1012:
751:
704:
628:
599:
425:
400:wave–particle duality
352:Augustin-Jean Fresnel
286:at infinity, and the
232:
131:
111:
103:
87:
79:
27:
3260:(3), AAPT: 243–247,
3217:(6), AIOP: 591–604,
3211:Computers in Physics
3170:Principles of optics
3076:. pp. 312–334.
2775:, SBN-10: 0198714742
2594:
2580:Fresnel zone concept
2450:
2430:
2410:
2390:
2363:
2336:
2202:
2175:
2007:
1967:Experimental aspects
1923:
1887:
1772:
1746:
1720:
1693:
1494:
1458:
1356:
1293:
1215:
1063:
1039:
818:
722:
637:
608:
454:
356:wave theory of light
344:Siméon Denis Poisson
187:
53:the picture tutorial
3499:1927Natur.119..558D
3455:1923Natur.112..540D
3411:2009PhRvA..79e3823R
3361:1922PhRv...20..594C
3312:2017NJPh...19c3022R
3266:1984AmJPh..52..243H
3223:1996ComPh..10..591D
3013:OEuvres Completes 1
2426:the wavelength and
2290:
2241:
2052:
1761:{\displaystyle r=t}
1735:{\displaystyle r=s}
1551:
1536:
1352:is hence given by:
925:
380:a century earlier.
336:experimentum crucis
305:and constructively
280:Newtonian telescope
177:Fresnel diffraction
158:Fresnel diffraction
2710:Other applications
2658:
2508:
2436:
2416:
2396:
2376:
2349:
2322:
2276:
2227:
2188:
2084:
2038:
1995:when close to the
1972:Intensity and size
1932:
1909:
1867:
1758:
1732:
1706:
1657:
1537:
1519:
1479:
1417:
1338:
1279:
1195:
1045:
1025:
1007:
911:
746:
699:
623:
594:
436:
311:geometrical optics
295:Huygens' principle
227:
134:
118:
106:
98:
82:
29:
3570:. 23 January 2015
2877:978-0-470-04472-8
2849:978-0-13-149933-1
2840:Pearson Education
2689:quantum mechanics
2647:
2645:
2506:
2460:
2439:{\displaystyle g}
2399:{\displaystyle R}
2316:
2267:
2212:
2185:
2120:is the wavelength
2078:
2001:plane wave source
1845:
1810:
1628:
1615:
1600:
1402:
1245:
1188:
1148:
1147:
1116:
975:
962:
947:
903:
856:
808:polar coordinates
744:
670:
570:
521:
385:quantum mechanics
216:
129:
74:
73:
3603:
3580:
3579:
3577:
3575:
3560:
3554:
3553:
3551:
3549:
3535:
3529:
3524:
3518:
3517:
3507:10.1038/119558a0
3482:
3476:
3475:
3466:
3464:10.1038/112540a0
3438:
3432:
3431:
3430:
3396:
3383:
3372:
3371:
3340:
3334:
3333:
3323:
3291:
3282:
3281:
3276:, archived from
3245:
3236:
3235:
3234:
3232:10.1063/1.168584
3206:
3200:
3194:
3188:
3187:
3164:
3158:
3157:
3146:
3136:
3127:
3121:
3109:
3103:
3101:
3089:
3083:
3082:
3074:Œuvres complètes
3069:
3063:
3055:
3039:
3033:
3024:
3018:
3017:
3004:
2998:
2997:
2987:
2981:
2980:
2961:
2955:
2954:
2951:"Poisson's Spot"
2947:
2941:
2940:
2923:
2917:
2916:
2898:(2nd ed.),
2897:
2887:
2881:
2880:
2866:(8th ed.),
2859:
2853:
2852:
2831:
2825:
2811:
2805:
2804:
2790:(1st ed.),
2783:
2777:
2776:
2754:
2719:. Finally, the
2667:
2665:
2664:
2659:
2648:
2646:
2644:
2633:
2625:
2617:
2616:
2607:
2571:
2566:
2561:
2553:
2517:
2515:
2514:
2509:
2507:
2505:
2498:
2497:
2487:
2476:
2462:
2461:
2458:
2445:
2443:
2442:
2437:
2425:
2423:
2422:
2417:
2405:
2403:
2402:
2397:
2385:
2383:
2382:
2377:
2375:
2374:
2358:
2356:
2355:
2350:
2348:
2347:
2331:
2329:
2328:
2323:
2321:
2317:
2315:
2307:
2296:
2289:
2284:
2272:
2268:
2266:
2258:
2247:
2240:
2235:
2214:
2213:
2210:
2197:
2195:
2194:
2189:
2187:
2186:
2183:
2147:
2142:
2137:
2093:
2091:
2090:
2085:
2083:
2079:
2077:
2069:
2058:
2051:
2046:
2025:
2024:
1959:
1954:
1949:
1941:
1939:
1938:
1933:
1918:
1916:
1915:
1910:
1905:
1904:
1876:
1874:
1873:
1868:
1863:
1862:
1861:
1860:
1851:
1846:
1838:
1828:
1827:
1826:
1825:
1816:
1811:
1803:
1790:
1789:
1767:
1765:
1764:
1759:
1741:
1739:
1738:
1733:
1715:
1713:
1712:
1707:
1705:
1704:
1671:trapezoidal rule
1666:
1664:
1663:
1658:
1636:
1635:
1634:
1630:
1629:
1621:
1616:
1608:
1601:
1596:
1595:
1594:
1582:
1576:
1550:
1545:
1535:
1527:
1512:
1511:
1488:
1486:
1485:
1480:
1426:
1424:
1423:
1418:
1413:
1412:
1403:
1401:
1400:
1399:
1387:
1386:
1376:
1375:
1366:
1347:
1345:
1344:
1339:
1337:
1336:
1331:
1327:
1323:
1322:
1288:
1286:
1285:
1280:
1278:
1277:
1272:
1268:
1267:
1266:
1259:
1246:
1238:
1227:
1226:
1204:
1202:
1201:
1196:
1191:
1190:
1189:
1187:
1186:
1174:
1173:
1164:
1159:
1149:
1146:
1145:
1133:
1132:
1123:
1119:
1117:
1112:
1111:
1110:
1103:
1089:
1081:
1080:
1054:
1052:
1051:
1046:
1016:
1014:
1013:
1008:
993:
992:
991:
990:
981:
977:
976:
968:
963:
955:
948:
940:
935:
924:
919:
904:
902:
894:
893:
892:
873:
859:
857:
852:
847:
836:
835:
755:
753:
752:
747:
745:
740:
732:
708:
706:
705:
700:
671:
669:
661:
656:
632:
630:
629:
624:
603:
601:
600:
595:
571:
569:
568:
559:
558:
557:
556:
544:
534:
532:
531:
522:
520:
519:
510:
509:
508:
507:
506:
494:
480:
472:
471:
288:secondary mirror
255:
249:
243:
236:
234:
233:
228:
217:
215:
207:
206:
197:
130:
115:
94:geometric optics
69:
66:
60:
26:
19:
3613:
3612:
3606:
3605:
3604:
3602:
3601:
3600:
3586:
3585:
3584:
3583:
3573:
3571:
3568:CU Bolder Today
3562:
3561:
3557:
3547:
3545:
3537:
3536:
3532:
3525:
3521:
3483:
3479:
3439:
3435:
3394:
3384:
3375:
3341:
3337:
3292:
3285:
3274:10.1119/1.13681
3246:
3239:
3207:
3203:
3197:Sommerfeld 1978
3195:
3191:
3185:
3165:
3161:
3155:
3137:
3130:
3110:
3106:
3090:
3086:
3070:
3066:
3040:
3036:
3025:
3021:
3005:
3001:
2988:
2984:
2978:
2962:
2958:
2949:
2948:
2944:
2938:
2924:
2920:
2914:
2888:
2884:
2878:
2870:, p. 992,
2860:
2856:
2850:
2842:, p. 315,
2832:
2828:
2812:
2808:
2802:
2794:, p. 374,
2784:
2780:
2773:
2765:, p. 444,
2755:
2751:
2746:
2729:
2712:
2677:
2634:
2626:
2624:
2612:
2608:
2606:
2595:
2592:
2591:
2544:
2540:
2523:
2493:
2489:
2488:
2477:
2475:
2457:
2453:
2451:
2448:
2447:
2431:
2428:
2427:
2411:
2408:
2407:
2391:
2388:
2387:
2370:
2366:
2364:
2361:
2360:
2343:
2339:
2337:
2334:
2333:
2308:
2297:
2295:
2291:
2285:
2280:
2259:
2248:
2246:
2242:
2236:
2231:
2209:
2205:
2203:
2200:
2199:
2182:
2178:
2176:
2173:
2172:
2156:
2107:
2070:
2059:
2057:
2053:
2047:
2042:
2020:
2016:
2008:
2005:
2004:
1974:
1969:
1924:
1921:
1920:
1900:
1896:
1888:
1885:
1884:
1856:
1852:
1847:
1837:
1836:
1832:
1821:
1817:
1812:
1802:
1801:
1797:
1785:
1781:
1773:
1770:
1769:
1747:
1744:
1743:
1721:
1718:
1717:
1700:
1696:
1694:
1691:
1690:
1688:
1684:
1620:
1607:
1606:
1602:
1590:
1586:
1578:
1577:
1575:
1574:
1570:
1546:
1541:
1528:
1523:
1507:
1503:
1495:
1492:
1491:
1459:
1456:
1455:
1451:
1408:
1404:
1395:
1391:
1382:
1378:
1377:
1371:
1367:
1365:
1357:
1354:
1353:
1332:
1318:
1314:
1307:
1303:
1302:
1294:
1291:
1290:
1273:
1255:
1254:
1250:
1237:
1236:
1232:
1231:
1222:
1218:
1216:
1213:
1212:
1182:
1178:
1169:
1165:
1163:
1155:
1154:
1150:
1141:
1137:
1128:
1124:
1118:
1099:
1098:
1094:
1090:
1088:
1076:
1072:
1064:
1061:
1060:
1058:
1040:
1037:
1036:
986:
982:
967:
954:
953:
949:
939:
931:
930:
926:
920:
915:
895:
869:
868:
864:
860:
858:
848:
846:
831:
827:
819:
816:
815:
805:
797:
782:
775:
733:
731:
723:
720:
719:
662:
657:
655:
638:
635:
634:
609:
606:
605:
564:
560:
552:
548:
540:
539:
535:
533:
527:
523:
515:
511:
502:
498:
490:
489:
485:
481:
479:
467:
463:
455:
452:
451:
449:
433:
429:
420:
394:Annus Mirabilis
389:Albert Einstein
323:
253:
247:
241:
208:
202:
198:
196:
188:
185:
184:
120:
113:
70:
64:
61:
50:
38:image placement
30:
24:
17:
12:
11:
5:
3611:
3610:
3599:
3598:
3582:
3581:
3555:
3530:
3519:
3477:
3433:
3373:
3335:
3283:
3237:
3201:
3189:
3183:
3159:
3153:
3128:
3122:From p. 140:
3104:
3084:
3064:
3042:Arago (1819).
3034:
3019:
2999:
2982:
2976:
2956:
2942:
2936:
2918:
2912:
2882:
2876:
2854:
2848:
2826:
2806:
2800:
2792:Addison Wesley
2778:
2772:978-0198714743
2771:
2748:
2747:
2745:
2742:
2741:
2740:
2738:Occulting disk
2735:
2728:
2725:
2711:
2708:
2676:
2673:
2657:
2654:
2651:
2643:
2640:
2637:
2632:
2629:
2623:
2620:
2615:
2611:
2605:
2602:
2599:
2539:
2536:
2522:
2519:
2504:
2501:
2496:
2492:
2486:
2483:
2480:
2474:
2471:
2468:
2465:
2456:
2435:
2415:
2395:
2373:
2369:
2346:
2342:
2320:
2314:
2311:
2306:
2303:
2300:
2294:
2288:
2283:
2279:
2275:
2271:
2265:
2262:
2257:
2254:
2251:
2245:
2239:
2234:
2230:
2226:
2223:
2220:
2217:
2208:
2181:
2155:
2152:
2128:
2127:
2121:
2115:
2109:
2105:
2082:
2076:
2073:
2068:
2065:
2062:
2056:
2050:
2045:
2041:
2037:
2034:
2031:
2028:
2023:
2019:
2015:
2012:
1973:
1970:
1968:
1965:
1931:
1928:
1908:
1903:
1899:
1895:
1892:
1866:
1859:
1855:
1850:
1844:
1841:
1835:
1831:
1824:
1820:
1815:
1809:
1806:
1800:
1796:
1793:
1788:
1784:
1780:
1777:
1757:
1754:
1751:
1731:
1728:
1725:
1703:
1699:
1686:
1682:
1679:Fresnel number
1675:Simpson's rule
1656:
1653:
1650:
1646:
1643:
1639:
1633:
1627:
1624:
1619:
1614:
1611:
1605:
1599:
1593:
1589:
1585:
1581:
1573:
1569:
1566:
1563:
1560:
1557:
1554:
1549:
1544:
1540:
1534:
1531:
1526:
1522:
1518:
1515:
1510:
1506:
1502:
1499:
1478:
1475:
1472:
1469:
1466:
1463:
1450:
1447:
1416:
1411:
1407:
1398:
1394:
1390:
1385:
1381:
1374:
1370:
1364:
1361:
1335:
1330:
1326:
1321:
1317:
1313:
1310:
1306:
1301:
1298:
1276:
1271:
1265:
1262:
1258:
1253:
1249:
1244:
1241:
1235:
1230:
1225:
1221:
1194:
1185:
1181:
1177:
1172:
1168:
1162:
1158:
1153:
1144:
1140:
1136:
1131:
1127:
1122:
1115:
1109:
1106:
1102:
1097:
1093:
1087:
1084:
1079:
1075:
1071:
1068:
1056:
1044:
1006:
1003:
1000:
996:
989:
985:
980:
974:
971:
966:
961:
958:
952:
946:
943:
938:
934:
929:
923:
918:
914:
910:
907:
901:
898:
891:
888:
885:
882:
879:
876:
872:
867:
863:
855:
851:
845:
842:
839:
834:
830:
826:
823:
803:
795:
780:
773:
767:
766:
760:
743:
739:
736:
730:
727:
717:
698:
695:
692:
689:
686:
683:
680:
677:
674:
668:
665:
660:
654:
651:
648:
645:
642:
622:
619:
616:
613:
593:
590:
587:
583:
580:
577:
574:
567:
563:
555:
551:
547:
543:
538:
530:
526:
518:
514:
505:
501:
497:
493:
488:
484:
478:
475:
470:
466:
462:
459:
447:
431:
427:
419:
416:
346:. In 1818 the
329:published his
322:
319:
258:
257:
251:
245:
226:
223:
220:
214:
211:
205:
201:
195:
192:
181:Fresnel number
179:. Namely, the
114:R = 5 μm = 10λ
72:
71:
65:September 2023
33:
31:
22:
15:
9:
6:
4:
3:
2:
3609:
3608:
3597:
3594:
3593:
3591:
3569:
3565:
3559:
3544:
3540:
3534:
3528:
3523:
3516:
3512:
3508:
3504:
3500:
3496:
3493:(2998): 558,
3492:
3488:
3481:
3474:
3470:
3465:
3460:
3456:
3452:
3449:(2815): 540,
3448:
3444:
3437:
3429:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3393:
3389:
3382:
3380:
3378:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3339:
3331:
3327:
3322:
3317:
3313:
3309:
3306:(3): 033022.
3305:
3301:
3297:
3290:
3288:
3280:on 2013-02-23
3279:
3275:
3271:
3267:
3263:
3259:
3255:
3251:
3244:
3242:
3233:
3228:
3224:
3220:
3216:
3212:
3205:
3199:, p. 186
3198:
3193:
3186:
3184:0-521-64222-1
3180:
3176:
3172:
3171:
3163:
3156:
3154:3-87144-377-8
3150:
3145:
3144:
3135:
3133:
3125:
3119:
3115:
3108:
3099:
3095:
3088:
3080:
3075:
3068:
3061:
3058:
3053:
3049:
3045:
3038:
3032:
3028:
3023:
3015:
3014:
3009:
3008:Fresnel, A.J.
3003:
2995:
2994:
2986:
2979:
2977:9780384704060
2973:
2969:
2968:
2960:
2952:
2946:
2939:
2937:0-321-18878-0
2933:
2929:
2922:
2915:
2913:0-393-95786-1
2909:
2905:
2901:
2896:
2895:
2886:
2879:
2873:
2869:
2865:
2858:
2851:
2845:
2841:
2837:
2830:
2824:
2820:
2816:
2810:
2803:
2801:0-201-02835-2
2797:
2793:
2789:
2782:
2774:
2768:
2764:
2760:
2753:
2749:
2739:
2736:
2734:
2731:
2730:
2724:
2722:
2718:
2707:
2704:
2702:
2698:
2694:
2690:
2686:
2682:
2672:
2668:
2655:
2652:
2649:
2641:
2638:
2635:
2630:
2627:
2621:
2618:
2613:
2609:
2603:
2600:
2588:
2585:
2581:
2576:
2572:
2570:
2565:
2560:
2555:
2551:
2547:
2535:
2533:
2529:
2518:
2502:
2499:
2494:
2490:
2484:
2481:
2478:
2472:
2466:
2454:
2433:
2413:
2393:
2371:
2367:
2344:
2340:
2318:
2312:
2309:
2304:
2301:
2298:
2292:
2286:
2281:
2277:
2273:
2269:
2263:
2260:
2255:
2252:
2249:
2243:
2237:
2232:
2228:
2224:
2218:
2206:
2179:
2170:
2166:
2162:
2151:
2148:
2146:
2141:
2136:
2131:
2125:
2122:
2119:
2116:
2113:
2110:
2104:
2100:
2097:
2096:
2095:
2080:
2074:
2071:
2066:
2063:
2060:
2054:
2048:
2043:
2039:
2035:
2029:
2026:
2021:
2017:
2010:
2002:
1998:
1994:
1989:
1987:
1983:
1979:
1976:For an ideal
1964:
1960:
1958:
1953:
1948:
1943:
1929:
1926:
1901:
1897:
1890:
1882:
1877:
1864:
1857:
1853:
1842:
1839:
1833:
1829:
1822:
1818:
1807:
1804:
1798:
1794:
1786:
1782:
1775:
1755:
1752:
1749:
1729:
1726:
1723:
1701:
1697:
1680:
1676:
1672:
1667:
1654:
1651:
1648:
1644:
1641:
1637:
1631:
1625:
1622:
1617:
1612:
1609:
1603:
1597:
1591:
1587:
1583:
1571:
1564:
1561:
1558:
1552:
1542:
1538:
1532:
1529:
1524:
1520:
1516:
1508:
1504:
1497:
1473:
1470:
1467:
1461:
1446:
1444:
1440:
1436:
1432:
1427:
1414:
1409:
1405:
1396:
1392:
1388:
1383:
1379:
1372:
1368:
1362:
1359:
1351:
1333:
1328:
1319:
1315:
1308:
1304:
1299:
1296:
1274:
1269:
1263:
1260:
1251:
1247:
1242:
1239:
1233:
1228:
1223:
1219:
1210:
1205:
1192:
1183:
1179:
1175:
1170:
1166:
1160:
1151:
1142:
1138:
1134:
1129:
1125:
1120:
1113:
1107:
1104:
1095:
1091:
1085:
1077:
1073:
1066:
1042:
1034:
1031:is large and
1030:
1021:
1017:
1004:
1001:
998:
994:
987:
983:
978:
972:
969:
964:
959:
956:
950:
944:
941:
936:
927:
916:
912:
908:
905:
899:
896:
886:
883:
880:
874:
865:
861:
853:
843:
840:
832:
828:
821:
813:
809:
801:
793:
789:
784:
779:
772:
764:
761:
759:
741:
737:
734:
728:
725:
718:
715:
712:
711:
710:
690:
684:
681:
678:
675:
666:
663:
652:
646:
640:
617:
611:
591:
588:
585:
578:
572:
565:
561:
553:
549:
545:
536:
528:
524:
516:
512:
503:
499:
495:
486:
482:
476:
468:
464:
457:
445:
441:
424:
415:
413:
409:
405:
401:
397:
395:
390:
386:
381:
379:
375:
370:
368:
363:
359:
357:
353:
349:
345:
340:
338:
337:
332:
328:
318:
316:
312:
308:
304:
300:
299:circumference
296:
291:
289:
285:
281:
277:
273:
268:
266:
265:laser sources
261:
252:
246:
240:
239:
238:
224:
221:
218:
212:
209:
203:
199:
193:
190:
183:must satisfy
182:
178:
174:
169:
167:
163:
159:
155:
151:
147:
143:
139:
110:
102:
95:
91:
86:
78:
68:
58:
54:
49:
47:
43:
39:
32:
21:
20:
3572:. Retrieved
3567:
3558:
3546:. Retrieved
3542:
3533:
3527:Feier et al.
3522:
3490:
3486:
3480:
3446:
3442:
3436:
3428:1721.1/51340
3402:
3399:Phys. Rev. A
3398:
3388:Holst, Bodil
3348:
3344:
3338:
3303:
3299:
3278:the original
3257:
3253:
3214:
3210:
3204:
3192:
3168:
3162:
3142:
3123:
3117:
3107:
3097:
3087:
3077:
3073:
3067:
3059:
3051:
3047:
3037:
3027:Fresnel 1868
3022:
3012:
3002:
2992:
2985:
2966:
2959:
2945:
2927:
2921:
2893:
2885:
2863:
2857:
2835:
2829:
2814:
2809:
2787:
2781:
2758:
2752:
2713:
2705:
2693:de Broglie's
2685:matter waves
2678:
2669:
2589:
2577:
2573:
2556:
2549:
2545:
2541:
2524:
2168:
2164:
2160:
2157:
2149:
2132:
2129:
2123:
2117:
2111:
2102:
2098:
1999:and using a
1997:optical axis
1990:
1978:point source
1975:
1961:
1944:
1880:
1878:
1668:
1452:
1442:
1438:
1434:
1430:
1428:
1349:
1206:
1032:
1028:
1026:
811:
806:). Going to
799:
791:
785:
777:
770:
768:
762:
713:
443:
437:
412:interference
393:
382:
371:
364:
360:
341:
334:
327:Thomas Young
324:
292:
284:point source
269:
262:
259:
170:
150:Fresnel spot
149:
146:Poisson spot
145:
141:
135:
62:
35:
3596:Diffraction
3355:: 594–600,
3057:From p. 16:
2900:W.W. Norton
2717:aberrations
1207:The source
408:diffraction
162:wave nature
90:diffraction
51:Please see
3574:9 February
3548:9 February
3345:Phys. Rev.
3029:, p.
2902:, p.
2823:1305537203
2744:References
2733:Aragoscope
2721:aragoscope
2552:= 1 m
1986:wavelength
1982:wave front
758:wavenumber
307:interferes
173:laser beam
142:Arago spot
42:formatting
3330:1367-2630
2681:deuterium
2650:−
2622:λ
2604:≈
2598:Δ
2584:interfere
2491:π
2485:λ
2473:≈
2414:λ
2313:λ
2305:π
2264:λ
2256:π
2072:λ
2061:π
2036:∝
1930:π
1898:θ
1840:π
1830:−
1805:π
1795:∝
1783:θ
1698:θ
1652:θ
1645:ρ
1638:ρ
1598:λ
1588:ρ
1584:π
1565:θ
1548:∞
1539:∫
1533:π
1521:∫
1517:∝
1474:θ
1209:intensity
1043:χ
922:∞
913:∫
909:π
854:λ
844:−
742:λ
738:π
691:χ
685:
667:λ
647:χ
618:χ
579:χ
525:∫
272:astronomy
219:≳
213:λ
210:ℓ
3590:Category
3390:(2009),
3010:(1868),
2727:See also
3515:4104602
3495:Bibcode
3473:4082518
3451:Bibcode
3407:Bibcode
3357:Bibcode
3308:Bibcode
3262:Bibcode
3219:Bibcode
3054:: 5–30.
2894:Physics
2528:evolute
756:is the
378:Maraldi
374:Delisle
321:History
156:due to
3513:
3487:Nature
3471:
3443:Nature
3328:
3181:
3151:
2974:
2934:
2928:Optics
2910:
2874:
2846:
2821:
2798:
2788:Optics
2769:
2332:where
2094:where
798:C and
418:Theory
404:photon
396:papers
237:where
154:shadow
140:, the
138:optics
44:, and
3511:S2CID
3469:S2CID
3395:(PDF)
3351:(6),
303:phase
278:in a
166:light
148:, or
3576:2017
3550:2017
3543:NASA
3326:ISSN
3179:ISBN
3149:ISBN
2972:ISBN
2932:ISBN
2908:ISBN
2872:ISBN
2844:ISBN
2819:ISBN
2796:ISBN
2767:ISBN
2359:and
1768:is:
1439:size
802:= CP
709:and
410:and
376:and
313:and
276:star
55:and
46:size
3503:doi
3491:119
3459:doi
3447:112
3423:hdl
3415:doi
3365:doi
3316:doi
3270:doi
3227:doi
3031:369
2904:984
2554:):
2459:rel
2211:rel
2184:rel
1742:to
1673:or
1443:b/a
794:= P
682:cos
391:'s
270:In
164:of
136:In
3592::
3566:.
3541:.
3509:,
3501:,
3489:,
3467:,
3457:,
3445:,
3421:,
3413:,
3403:79
3401:,
3397:,
3376:^
3363:,
3349:20
3347:,
3324:.
3314:.
3304:19
3302:.
3298:.
3286:^
3268:,
3258:52
3256:,
3252:,
3240:^
3225:,
3215:10
3213:,
3177:,
3131:^
3052:11
2906:,
2761:,
2548:=
2165:Sb
783:.
358:.
339:.
144:,
40:,
3578:.
3552:.
3505::
3497::
3461::
3453::
3425::
3417::
3409::
3367::
3359::
3332:.
3318::
3310::
3272::
3264::
3229::
3221::
2953:.
2656:.
2653:r
2642:b
2639:+
2636:g
2631:b
2628:g
2619:+
2614:2
2610:r
2601:r
2550:b
2546:g
2503:R
2500:w
2495:2
2482:g
2479:2
2470:)
2467:w
2464:(
2455:I
2434:g
2394:R
2372:1
2368:J
2345:0
2341:J
2319:)
2310:g
2302:R
2299:w
2293:(
2287:2
2282:1
2278:J
2274:+
2270:)
2261:g
2253:R
2250:w
2244:(
2238:2
2233:0
2229:J
2225:=
2222:)
2219:w
2216:(
2207:I
2180:I
2169:g
2167:/
2161:S
2124:b
2118:λ
2112:d
2106:1
2103:P
2099:r
2081:)
2075:b
2067:d
2064:r
2055:(
2049:2
2044:0
2040:J
2033:)
2030:r
2027:,
2022:1
2018:P
2014:(
2011:U
1927:2
1907:)
1902:1
1894:(
1891:I
1881:s
1865:.
1858:2
1854:t
1849:i
1843:2
1834:e
1823:2
1819:s
1814:i
1808:2
1799:e
1792:)
1787:1
1779:(
1776:R
1756:t
1753:=
1750:r
1730:s
1727:=
1724:r
1702:1
1687:1
1685:P
1683:0
1655:.
1649:d
1642:d
1632:)
1626:b
1623:1
1618:+
1613:g
1610:1
1604:(
1592:2
1580:i
1572:e
1568:)
1562:,
1559:r
1556:(
1553:g
1543:0
1530:2
1525:0
1514:)
1509:1
1505:P
1501:(
1498:U
1477:)
1471:,
1468:r
1465:(
1462:g
1435:b
1431:b
1415:.
1410:0
1406:I
1397:2
1393:a
1389:+
1384:2
1380:b
1373:2
1369:b
1363:=
1360:I
1350:b
1334:2
1329:|
1325:)
1320:1
1316:P
1312:(
1309:U
1305:|
1300:=
1297:I
1275:2
1270:|
1264:g
1261:k
1257:i
1252:e
1248:A
1243:g
1240:1
1234:|
1229:=
1224:0
1220:I
1193:.
1184:2
1180:a
1176:+
1171:2
1167:b
1161:k
1157:i
1152:e
1143:2
1139:a
1135:+
1130:2
1126:b
1121:b
1114:g
1108:g
1105:k
1101:i
1096:e
1092:A
1086:=
1083:)
1078:1
1074:P
1070:(
1067:U
1057:1
1033:b
1029:g
1005:.
1002:r
999:d
995:r
988:2
984:r
979:)
973:b
970:1
965:+
960:g
957:1
951:(
945:2
942:1
937:k
933:i
928:e
917:a
906:2
900:b
897:g
890:)
887:b
884:+
881:g
878:(
875:k
871:i
866:e
862:A
850:i
841:=
838:)
833:1
829:P
825:(
822:U
812:a
804:1
800:b
796:0
792:g
781:1
778:r
774:0
771:r
763:S
735:2
729:=
726:k
714:A
697:)
694:)
688:(
679:+
676:1
673:(
664:2
659:i
653:=
650:)
644:(
641:K
621:)
615:(
612:K
592:,
589:S
586:d
582:)
576:(
573:K
566:1
562:r
554:1
550:r
546:k
542:i
537:e
529:S
517:0
513:r
504:0
500:r
496:k
492:i
487:e
483:A
477:=
474:)
469:1
465:P
461:(
458:U
448:1
444:E
434:.
432:0
428:1
254:λ
248:â„“
242:d
225:,
222:1
204:2
200:d
194:=
191:F
116:.
96:.
67:)
63:(
48:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
