504:
film, digital) used to observe the diffraction pattern can have an intensity threshold for detection, the full diffraction pattern may not be apparent. In astronomy, the outer rings are frequently not apparent even in a highly magnified image of a star. It may be that none of the rings are apparent, in which case the star image appears as a disk (central maximum only) rather than as a full diffraction pattern. Furthermore, fainter stars will appear as smaller disks than brighter stars, because less of their central maximum reaches the threshold of detection. While in theory all stars or other "point sources" of a given wavelength and seen through a given aperture have the same Airy disk radius characterized by the above equation (and the same diffraction pattern size), differing only in intensity, the appearance is that fainter sources appear as smaller disks, and brighter sources appear as larger disks. This was described by Airy in his original work:
523:
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60:
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1016:
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44:
3001:
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20:
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32:
159:...the star is then seen (in favourable circumstances of tranquil atmosphere, uniform temperature, etc.) as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders. They succeed each other nearly at equal intervals round the central disc....
185:. Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by such a lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be
4536:
509:
explained. Thus the radius of the spurious disk of a faint star, where light of less than half the intensity of the central light makes no impression on the eye, is determined by , whereas the radius of the spurious disk of a bright star, where light of 1/10 the intensity of the central light is sensible, is determined by .
4195:
3504:
Similar equations can also be derived for the obscured Airy diffraction pattern which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs
877:) require the user to align a peep sight (rear, nearby sight, i.e. which will be out of focus) with a tip (which should be focused and overlaid on the target) at the end of the barrel. When looking through the peep sight, the user will notice an Airy disk that will help center the sight over the pin.
526:
Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space
Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas
1006:
In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the
999:
In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to
508:
The rapid decrease of light in the successive rings will sufficiently explain the visibility of two or three rings with a very bright star and the non-visibility of rings with a faint star. The difference of the diameters of the central spots (or spurious disks) of different stars ... is also fully
503:
However, while the angle at which the first minimum occurs (which is sometimes described as the radius of the Airy disk) depends only on wavelength and aperture size, the appearance of the diffraction pattern will vary with the intensity (brightness) of the light source. Because any detector (eye,
499:
for barely resolving two objects that are point sources of light, such as stars seen through a telescope, is that the center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second. This means that the angular resolution of a diffraction-limited system is
4228:
The practical effect of having a central obstruction in a telescope is that the central disc becomes slightly smaller, and the first bright ring becomes brighter at the expense of the central disc. This becomes more problematic with short focal length telescopes which require larger secondary
4466:
A Gaussian beam transmitted through a hard aperture will be clipped. Energy is lost and edge diffraction occurs, effectively increasing the divergence. Because of these effects there is a
Gaussian beam diameter which maximizes the intensity in the far field. This occurs when the
3678:
1798:
513:
Despite this feature of Airy's work, the radius of the Airy disk is often given as being simply the angle of first minimum, even in standard textbooks. In reality, the angle of first minimum is a limiting value for the size of the Airy disk, and not a definite radius.
4881:{\displaystyle \int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{i{\vec {k}}\cdot {\vec {r}}}=\int _{0}^{a}rdr\int _{0}^{2\pi }d\phi e^{ikr\cos \phi }=2\int _{0}^{a}rdr\int _{0}^{\pi }d\phi \cos(kr\cos \phi )=2\pi \int _{0}^{a}rdrJ_{0}(kr)=2\pi {\frac {a}{k}}J_{1}(kr).}
3903:
536:
If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects cannot be clearly separated any more in the image, and they start blurring together. Two objects are said to be
892:
The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance
5116:
851:
with approximately 1 μm diameter. However, at this f-number, spherical aberration limits visual acuity, while a 3 mm pupil diameter (f/5.7) approximates the resolution achieved by the human eye. The maximum density of cones in the human
2068:
1929:
2533:
3139:
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A circular laser beam with uniform intensity across the circle (a flat-top beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.
403:
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239:
5207:
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The Airy pattern falls rather slowly to zero with increasing distance from the center, with the outer rings containing a significant portion of the integrated intensity of the pattern. As a result, the
648:
592:
282:
1283:
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3033:(RMS) spotsize is undefined (i.e. infinite). An alternative measure of the spot size is to ignore the relatively small outer rings of the Airy pattern and to approximate the central lobe with a
2676:
3745:
738:
3398:
3339:
1686:
1843:
2416:
856:
is approximately 170,000 per square millimeter, which implies that the cone spacing in the human eye is about 2.5 μm, approximately the diameter of the point spread function at f/5.
1000:
the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.
2194:
1966:
2953:
4223:
4237:
A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the lens. The intensity at the center of the focus will be
1384:
995:
491:
was the radius of the aperture in inches, and the wavelength of light was assumed to be 0.000022 inches (560 nm; the mean of visible wavelengths). This is equal to the
443:
4937:
2353:
2321:
2226:
1058:
3264:
4190:{\displaystyle E(R)={\frac {1}{(1-\epsilon ^{2})}}\left(1-J_{0}^{2}(x)-J_{1}^{2}(x)+\epsilon ^{2}\left-4\epsilon \int _{0}^{x}{\frac {J_{1}(t)J_{1}(\epsilon t)}{t}}\,dt\right)}
2289:
3042:
2839:
3495:
1525:
appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming
765:
4401:
3466:
2557:
1068:
Diffraction from an aperture with a lens. The far field image will (only) be formed at the screen one focal length away, where R=f (f=focal length). The observation angle
3171:
2593:
2114:
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3700:
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3852:
3440:
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3022:
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951:
324:
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3004:
A radial cross-section through the Airy pattern (solid curve) and its
Gaussian profile approximation (dashed curve). The abscissa is given in units of the wavelength
1424:
1086:
304:
4357:
2895:
2868:
2409:
2382:
1827:
1339:
1312:
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diameter of the
Gaussian is 89% of the aperture diameter, and the on axis intensity in the far field will be 81% of that produced by a uniform intensity profile.
885:
Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to
489:
467:
346:
4929:
4909:
4528:
4461:
4421:
3896:
3872:
3832:
3191:
1404:
931:
911:
810:
688:
668:
527:
the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though 20/20 vision resolves to only 60 arcsecs (1 arcminute)
1571:
197:
Far from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula:
832:
Longitudinal sections through a focused beam with (top) negative, (center) zero, and (bottom) positive spherical aberration. The lens is to the left.
166:
Mathematically, the diffraction pattern is characterized by the wavelength of light illuminating the circular aperture, and the aperture's size. The
163:
Airy wrote the first full theoretical treatment explaining the phenomenon (his 1835 "On the
Diffraction of an Object-glass with Circular Aperture").
202:
1426:
is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point.
5551:
Derenzo, S. E. (2023). "Feynman photon path integral calculations of optical reflection, diffraction, and scattering from conduction electrons".
606:
1429:
3415:). The Gaussian profile approximation provides an alternative means of comparison: using the approximation above shows that the Gaussian waist
551:
248:
1113:
3673:{\displaystyle I(R)={\frac {I_{0}}{(1-\epsilon ^{2})^{2}}}\left({\frac {2J_{1}(x)}{x}}-{\frac {2\epsilon J_{1}(\epsilon x)}{x}}\right)^{2}}
1793:{\displaystyle \sin \theta \approx {\frac {3.83}{ka}}={\frac {3.83\lambda }{2\pi a}}=1.22{\frac {\lambda }{2a}}=1.22{\frac {\lambda }{d}}.}
351:
5694:
Mahajan, Virendra N. (1986). "Uniform versus
Gaussian beams: a comparison of the effects of diffraction, osbscuration, and aberrations".
3750:
3702:
is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam).
697:
3352:. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes
913:
from the aperture where the diffracted light is observed (the screen distance) is large compared to the aperture size, and the radius
170:
of the diffraction pattern is additionally characterized by the sensitivity of the eye or other detector used to observe the pattern.
548:
Therefore, the smallest angular separation two objects can have before they significantly blur together is given as stated above by
816:
smaller than half this value (one pixel for each object, one for each space between) would not significantly increase the captured
5460:
5127:
3220:
is the
Gaussian RMS width (in one dimension). If we equate the peak amplitude of the Airy pattern and Gaussian profile, that is,
5365:"Objects in Telescope Are Farther Than They Appear – How diffraction tricked Galileo into mismeasuring distances to the stars"
690:
is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the
5880:
4240:
1007:
iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.
412:
5414:
2598:
3705:
1003:
Hence, the focal spot of a uniform circular laser beam (a flattop beam) focused by a lens will also be an Airy pattern.
5845:
2958:
The Airy disk and diffraction pattern can be computed numerically from first principles using
Feynman path integrals.
2715:
5474:
5442:
3878:
of the system. The fractional encircled energy (the fraction of the total energy contained within a circle of radius
3358:
3299:
5651:
Rivolta, Claudio (1986). "Airy disk diffraction pattern: comparison of some values of f/No. and obscuration ratio".
2710:
above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:
1645:
3510:
1506:
is its distance to the aperture. Note that the Airy disk as given by the above expression is only valid for large
2678:
The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.
2134:
600:. The larger the aperture for a given wavelength, the finer the detail that can be distinguished in the image.
542:
2904:
541:
when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the
5288:
Transactions
Treatises on physical astronomy, light and sound contributed to the Encyclopaedia Metropolitana
5111:{\displaystyle \int _{x^{2}/a^{2}+y^{2}/b^{2}\leq 1}e^{ik_{x}x}e^{ik_{y}y}dxdy=2\pi {\frac {ab}{c}}J_{1}(c)}
4202:
39:). Note that the red component is diffracted more than the blue, so that the center appears slightly bluish.
5824:
5764:
Borghi, Riccardo (2014). "Plane-wave
Fresnel diffraction by elliptic apertures: a Fourier-based approach".
5364:
1348:
956:
2901:. Hence the fractions of the total power contained within the first, second, and third dark rings (where
2326:
2294:
2199:
1022:
5814:
2231:
151:
1924:{\displaystyle q_{1}=R\sin \theta _{1}\approx 1.22{R}{\frac {\lambda }{d}}=1.22{\frac {\lambda }{2A}}}
2528:{\displaystyle I_{0}={\frac {\mathrm {E} _{A}^{2}A^{2}}{2R^{2}}}={\frac {P_{0}A}{\lambda ^{2}R^{2}}}}
3471:
743:
5927:
5466:
4362:
3445:
3411:
if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see
3223:
2121:
522:
2540:
2562:
2063:{\displaystyle A={\frac {r}{R'}}={\frac {r}{\sqrt {f^{2}+r^{2}}}}={\frac {1}{\sqrt {4N^{2}+1}}};}
4891:
This is the special case of the Fourier integral of the elliptical cross section with half axes
2684:
2079:
59:
27:
intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern.
5597:
5228:
3685:
1949:/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance
1511:
1097:
886:
125:
The diffraction pattern resulting from a uniformly illuminated, circular aperture has a bright
4426:
3837:
3007:
1535:
936:
309:
5248:
5233:
4470:
3442:
of the Gaussian approximation to the Airy disk is about two-third the Airy disk radius, i.e.
3418:
3269:
3196:
2983:
1409:
1071:
848:
289:
3146:
5932:
5773:
5703:
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1805:
1317:
1290:
8:
5895:
5810:
5490:
5283:
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3412:
3134:{\displaystyle I(q)\approx I'_{0}\exp \left({\frac {-2q^{2}}{\omega _{0}^{2}}}\right)\ ,}
1515:
1093:
472:
450:
329:
186:
134:
126:
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1642:
From this, it follows that the first dark ring in the diffraction pattern occurs where
1389:
916:
896:
795:
673:
653:
496:
492:
348:
is the diameter of the aperture in meters. The full width at half maximum is given by
130:
820:. However, it may improve the final image by over-sampling, allowing noise reduction.
5892:
5789:
5676:
5633:
5625:
5470:
5438:
5398:
5332:
5302:
5238:
5218:
3034:
2559:
is the source strength per unit area at the aperture, A is the area of the aperture (
1105:
1635:{\displaystyle x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots .}
784:). For violet, the shortest wavelength visible light, the wavelength λ is about 420
5781:
5711:
5668:
5617:
5578:
5376:
3030:
817:
5730:
1802:
If a lens is used to focus the Airy pattern at a finite distance, then the radius
1514:
applies; calculation of the shadow in the near-field must rather be handled using
5598:"Gaussian approximations of fluorescence microscope point-spread function models"
5348:
5263:
2898:
1342:
1101:
853:
91:
5877:(website), November 8, 2005. – Mathematical details to derive the above formula.
5754:
A.E. Siegman, Lasers, Se. 18.4, University Science Books, Mill Valley, CA, 1989
5582:
5434:
596:
Thus, the ability of the system to resolve detail is limited by the ratio of λ/
88:
36:
5843:
5822:
1064:
1015:
5921:
5629:
5397:
Airy, G. B., "On the Diffraction of an Object-glass with Circular Aperture",
5331:
Airy, G. B., "On the Diffraction of an Object-glass with Circular Aperture",
828:
781:
138:
99:
5785:
5793:
5715:
5680:
5637:
5258:
5243:
813:
691:
43:
1941:/2 divided by R', the distance from the center of the Airy pattern to the
1502:
is the radial distance from the observation point to the optical axis and
1491:{\displaystyle x=ka\sin \theta ={\frac {2\pi a}{\lambda }}{\frac {q}{R}},}
1019:
Diffraction from a circular aperture. The Airy pattern is observable when
5881:"The Airy Disk: An Explanation Of What It Is, And Why You Can't Avoid It"
5672:
5621:
5223:
2595:) and R is the distance from the aperture. At the focal plane of a lens,
2125:
107:
64:
5868:
5596:
Zhang, Bo; Zerubia, Josiane; Olivo-Marin, Jean-Christophe (2007-04-01).
2384:
at the center of the diffraction pattern is related to the total power
1526:
837:
178:
52:
5380:
5900:
844:
789:
785:
398:{\displaystyle \theta _{\mathrm {FWHM} }=1.029{\frac {\lambda }{d}}.}
182:
146:
119:
24:
670:
is the separation of the images of the two objects on the film, and
5565:
3898:
centered at the optical axis in the focal plane) is then given by:
3875:
3807:{\displaystyle x=ka\sin(\theta )\approx {\frac {\pi R}{\lambda N}}}
3349:
3193:
represents the radial distance from the center of the pattern, and
1837:
840:
772:
768:
103:
5911:"Extended Nijboer-Zernike (ENZ) Analysis and Aberration Retrieval"
812:
of about 4 μm. In a digital camera, making the pixels of the
5731:"Chapter 7 Obstruction effects (7.1. Central obstruction effect)"
3834:
is the radial distance in the focal plane from the optical axis,
2124:
by the numerical aperture (and thus f-number) of its lens due to
1829:
of the first dark ring on the focal plane is solely given by the
1314:
is the maximum intensity of the pattern at the Airy disc center,
953:
of the light. The last two conditions can be formally written as
771:
of a lens. A typical setting for use on an overcast day would be
111:
5384:
31:
5910:
3000:
2965:
874:
174:
115:
72:
19:
5890:
5303:"On the Diffraction of an Object-glass with Circular Aperture"
1961:(ratio of the focal length to the lens diameter) according to
1529:
light incident on the aperture) given by the above equations.
149:
under high magnification for an 1828 article on light for the
137:. The disk and rings phenomenon had been known prior to Airy;
133:
rings around is called the Airy pattern. Both are named after
5844:
Kenneth R. Spring; Brian O. Flynn & Michael W. Davidson.
5823:
Kenneth R. Spring; Brian O. Flynn & Michael W. Davidson.
4510:
The Fourier integral of the circular cross section of radius
234:{\displaystyle \sin \theta \approx 1.22{\frac {\lambda }{d}}}
95:
48:
5202:{\displaystyle c\equiv {\sqrt {(k_{x}a)^{2}+(k_{y}b)^{2}}}.}
129:, known as the Airy disk, which together with the series of
35:
A computer-generated Airy disk from diffracted white light (
3293:
giving the optimal approximation to the pattern, we obtain
933:
of the aperture is not too much larger than the wavelength
643:{\displaystyle {\frac {x}{f}}=1.22\,{\frac {\lambda }{d}},}
142:
5825:"Image Formation: Numerical Aperture and Image Resolution"
5510:
2995:
792:
for sensitivity of S cone cells). This gives a value for
587:{\displaystyle \sin \theta =1.22\,{\frac {\lambda }{d}}.}
277:{\displaystyle \theta \approx 1.22{\frac {\lambda }{d}},}
1278:{\displaystyle I(\theta )=I_{0}\left^{2}=I_{0}\left^{2}}
5595:
4325:{\displaystyle I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}
4225:
the formulas reduce to the unobscured versions above.
3421:
3407:
theory, it is common to describe an imaging system as
3361:
3302:
3272:
3199:
2082:
5130:
4940:
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is about 2.1, corresponding to a diffraction-limited
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609:
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475:
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332:
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173:
The most important application of this concept is in
4232:
5400:
Transactions of the Cambridge Philosophical Society
5334:
Transactions of the Cambridge Philosophical Society
5307:
Transactions of the Cambridge Philosophical Society
2671:{\displaystyle I_{0}=(P_{0}A)/(\lambda ^{2}f^{2}).}
5416:Physics for Scientists and Engineers (3rd edition)
5201:
5110:
4923:
4903:
4880:
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4395:
4351:
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3740:{\displaystyle \left(0\leq \epsilon <1\right),}
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3333:
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3258:
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2987:The encircled power graphed next to the intensity.
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483:
469:was the angle of first minimum in seconds of arc,
461:
437:
397:
340:
318:
298:
276:
233:
5815:"Concepts and Formulas in Microscopy: Resolution"
2131:The half maximum of the central Airy disk (where
733:{\displaystyle x=1.22\,{\frac {\lambda \,f}{d}},}
5919:
3393:{\textstyle \omega _{0}\approx 0.90\lambda N\ .}
3334:{\textstyle \omega _{0}\approx 0.84\lambda N\ ,}
3173:is the irradiance at the center of the pattern,
1945:of the aperture. Viewing the aperture of radius
23:A computer-generated image of an Airy disk. The
5809:
3505:that incorporate a secondary mirror, including
1681:{\displaystyle ka\sin {\theta }=3.8317\dots ,}
326:is the wavelength of the light in meters, and
55:pinhole aperture with 27 orders of diffraction
5458:
1100:pattern of a circular aperture, given by the
880:
2955:) are 83.8%, 91.0%, and 93.8% respectively.
2323:and the maximum of the first ring occurs at
110:of light. The Airy disk is of importance in
1010:
1957:is related to the commonly-cited f-number
67:at f/25 aperture. Image size: 1×1 mm.
47:A real Airy disk created by passing a red
5846:"Image Formation: Airy Pattern Formation"
5564:
4175:
3024:times the f-number of the optical system.
2189:{\displaystyle 2J_{1}(x)/x=1/{\sqrt {2}}}
1190:
1169:
717:
710:
626:
570:
5290:. Richard Griffin & Co. p. 491.
5281:
3499:
2999:
2982:
2964:
2116:This shows that the best possible image
1063:
1014:
827:
521:
58:
42:
30:
18:
5866:
5693:
5687:
5650:
5550:
5269:
2948:{\displaystyle J_{1}(ka\sin \theta )=0}
1088:stays the same as in the lensless case.
5920:
5869:"Diffraction from a Circular Aperture"
5763:
5362:
5353:, Dover Publications, 1980, pp. 39–40.
4505:
4218:{\displaystyle \epsilon \rightarrow 0}
2996:Approximation using a Gaussian profile
5891:
5728:
5428:
859:
141:described the appearance of a bright
5735:7. Notes on amateur telescope optics
5511:http://en.wikibooks.org/Marksmanship
5300:
1379:{\displaystyle k={2\pi }/{\lambda }}
990:{\displaystyle R>a^{2}/\lambda .}
438:{\displaystyle s={\frac {2.76}{a}},}
2348:{\displaystyle x=5.13562230\dots .}
2316:{\displaystyle x=2.58383899\dots ,}
2221:{\displaystyle x=1.61633995\dots ;}
1406:is the radius of the aperture, and
1053:{\displaystyle R\gg a^{2}/\lambda }
63:Airy disk captured by 2000 mm
13:
2545:
2438:
1937:is equal to the aperture's radius
370:
367:
364:
361:
14:
5944:
5803:
5729:Sacek, Vladimir (July 14, 2006).
5553:Nuclear Instruments and Methods A
4463:is the focal length of the lens.
4233:Comparison to Gaussian beam focus
2969:The Airy Pattern on the interval
2284:{\displaystyle 2J_{1}(x)/x=1/{e}}
5541:(Pergamon Press, New York, 1965)
4359:is the total power of the beam,
2834:{\displaystyle P(\theta )=P_{0}}
2076:≫1 it is simply approximated as
1345:of the first kind of order one,
1096:of the Airy pattern follows the
873:Some weapon aiming sights (e.g.
823:
5757:
5748:
5722:
5644:
5589:
5544:
5529:
5516:
5503:
5483:
5363:Graney, Christopher M. (2009).
1521:However the exact Airy pattern
868:
5452:
5422:
5419:, Prentice-Hall, 2000, p. 896.
5407:
5391:
5356:
5341:
5325:
5294:
5275:
5185:
5168:
5156:
5139:
5105:
5099:
4872:
4863:
4831:
4822:
4776:
4758:
4617:
4602:
4319:
4296:
4288:
4272:
4209:
4166:
4157:
4144:
4138:
4093:
4084:
4063:
4054:
4012:
4006:
3985:
3979:
3947:
3928:
3916:
3910:
3778:
3772:
3650:
3641:
3610:
3604:
3570:
3550:
3531:
3525:
3490:{\displaystyle 1.22\lambda N.}
3055:
3049:
2936:
2918:
2828:
2825:
2807:
2786:
2768:
2744:
2728:
2722:
2697:
2691:
2662:
2639:
2631:
2615:
2254:
2248:
2157:
2151:
1555:
1549:
1256:
1250:
1182:
1163:
1126:
1120:
760:{\displaystyle {\frac {f}{d}}}
603:This can also be expressed as
1:
5819:Nikon MicroscopyU (website).
5350:Amateur Astronomer's Handbook
4396:{\displaystyle A=\pi D^{2}/4}
3511:Schmidt–Cassegrain telescopes
3461:{\displaystyle 0.84\lambda N}
3259:{\displaystyle I'_{0}=I_{0},}
1933:where the numerical aperture
495:of a circular aperture. The
243:or, for small angles, simply
87:are descriptions of the best-
16:Diffraction pattern in optics
5857:(Interactive Java Tutorial)
5836:(Interactive Java Tutorial)
2552:{\displaystyle \mathrm {E} }
2411:incident on the aperture by
500:given by the same formulae.
407:Airy wrote this relation as
7:
5459:Steve Chapman, ed. (2000).
5282:Herschel, J. F. W. (1828).
5212:
3747:and x is defined as above:
2588:{\displaystyle A=\pi a^{2}}
2109:{\textstyle A\approx 1/2N.}
517:
10:
5949:
5583:10.1016/j.nima.2023.168679
2703:{\displaystyle I(\theta )}
1953:, the numerical aperture
1108:of the circular aperture:
881:Conditions for observation
531:
152:Encyclopedia Metropolitana
4403:is the area of the beam (
3695:{\displaystyle \epsilon }
1836:(closely related to the
889:(far-field diffraction).
106:can make, limited by the
5467:McGraw-Hill Professional
4436:{\displaystyle \lambda }
3847:{\displaystyle \lambda }
3435:{\textstyle \omega _{0}}
3286:{\textstyle \omega _{0}}
3213:{\textstyle \omega _{0}}
3017:{\displaystyle \lambda }
1561:{\displaystyle J_{1}(x)}
1011:Mathematical formulation
946:{\displaystyle \lambda }
319:{\displaystyle \lambda }
5896:"Bessel Function Zeros"
5786:10.1364/JOSAA.31.002120
5526:, Addison Wesley (2001)
4495:{\displaystyle 1/e^{2}}
4443:is the wavelength, and
4423:is the beam diameter),
1419:{\displaystyle \theta }
1081:{\displaystyle \theta }
1060:(i.e. in the far field)
299:{\displaystyle \theta }
192:
5716:10.1364/JOSAA.3.000470
5491:"Eye Receptor Density"
5429:Hecht, Eugene (1987).
5229:Fraunhofer diffraction
5203:
5112:
4925:
4905:
4882:
4524:
4496:
4457:
4437:
4417:
4397:
4353:
4326:
4219:
4191:
3892:
3868:
3854:is the wavelength and
3848:
3828:
3808:
3741:
3696:
3674:
3491:
3462:
3436:
3394:
3335:
3287:
3266:and find the value of
3260:
3214:
3187:
3167:
3166:{\displaystyle I'_{0}}
3135:
3025:
3018:
2988:
2978:
2949:
2891:
2864:
2835:
2704:
2672:
2589:
2553:
2529:
2405:
2378:
2349:
2317:
2285:
2222:
2190:
2110:
2064:
1925:
1823:
1794:
1682:
1636:
1562:
1512:Fraunhofer diffraction
1492:
1420:
1400:
1380:
1335:
1308:
1279:
1098:Fraunhofer diffraction
1089:
1082:
1061:
1054:
991:
947:
927:
907:
887:Fraunhofer diffraction
833:
806:
761:
734:
684:
664:
644:
588:
528:
511:
485:
463:
439:
399:
342:
320:
300:
278:
235:
161:
68:
56:
40:
28:
5859:Molecular Expressions
5838:Molecular Expressions
5535:M. Born and E. Wolf,
5462:Optical System Design
5249:Point spread function
5234:Bloom (shader effect)
5204:
5113:
4926:
4906:
4883:
4525:
4497:
4458:
4438:
4418:
4398:
4354:
4352:{\displaystyle P_{0}}
4327:
4220:
4192:
3893:
3869:
3849:
3829:
3809:
3742:
3697:
3675:
3500:Obscured Airy pattern
3492:
3463:
3437:
3395:
3336:
3288:
3261:
3215:
3188:
3168:
3136:
3019:
3003:
2986:
2968:
2950:
2892:
2890:{\displaystyle J_{1}}
2865:
2863:{\displaystyle J_{0}}
2836:
2705:
2673:
2590:
2554:
2530:
2406:
2404:{\displaystyle P_{0}}
2379:
2377:{\displaystyle I_{0}}
2350:
2318:
2286:
2228:the 1/e point (where
2223:
2191:
2111:
2065:
1926:
1824:
1822:{\displaystyle q_{1}}
1795:
1683:
1637:
1563:
1493:
1421:
1401:
1381:
1336:
1334:{\displaystyle J_{1}}
1309:
1307:{\displaystyle I_{0}}
1280:
1083:
1067:
1055:
1018:
992:
948:
928:
908:
849:point spread function
831:
807:
762:
735:
694:of the lens, we find
685:
665:
645:
589:
525:
506:
486:
464:
440:
400:
343:
321:
301:
279:
236:
157:
62:
46:
34:
22:
5673:10.1364/AO.25.002404
5622:10.1364/AO.46.001819
5538:Principles of Optics
5301:Airy, G. B. (1835).
5270:Notes and references
5128:
4938:
4915:
4895:
4537:
4514:
4471:
4447:
4427:
4407:
4363:
4336:
4241:
4203:
3904:
3882:
3858:
3838:
3818:
3751:
3706:
3686:
3519:
3507:Newtonian telescopes
3472:
3446:
3419:
3359:
3300:
3270:
3224:
3197:
3177:
3147:
3043:
3008:
2905:
2874:
2847:
2716:
2685:
2599:
2563:
2541:
2417:
2388:
2361:
2327:
2295:
2232:
2200:
2135:
2080:
1967:
1844:
1806:
1694:
1646:
1572:
1536:
1430:
1410:
1390:
1349:
1318:
1291:
1114:
1072:
1023:
957:
937:
917:
897:
796:
744:
698:
674:
654:
607:
552:
473:
451:
413:
352:
330:
310:
290:
249:
203:
5811:Michael W. Davidson
5778:2014JOSAA..31.2120B
5708:1986JOSAA...3..470M
5665:1986ApOpt..25.2404R
5614:2007ApOpt..46.1819Z
5575:2023NIMPA105668679D
5513:, "Sight Alignment"
5369:The Physics Teacher
5319:1835TCaPS...5..283A
5254:Debye-Scherrer ring
4802:
4745:
4721:
4669:
4642:
4581:
4554:
4506:Elliptical aperture
4124:
4083:
4053:
4005:
3978:
3413:Optical lens design
3409:diffraction-limited
3239:
3162:
3118:
3073:
3037:profile, such that
2806:
2767:
2681:The expression for
2452:
1516:Fresnel diffraction
1386:is the wavenumber,
484:{\displaystyle {a}}
462:{\displaystyle {s}}
341:{\displaystyle {d}}
187:diffraction limited
135:George Biddell Airy
5893:Weisstein, Eric W.
5766:J. Opt. Soc. Am. A
5696:J. Opt. Soc. Am. A
5199:
5108:
4921:
4901:
4878:
4788:
4731:
4707:
4652:
4628:
4564:
4540:
4520:
4492:
4453:
4433:
4413:
4393:
4349:
4322:
4215:
4187:
4110:
4069:
4039:
3991:
3964:
3888:
3864:
3844:
3824:
3804:
3737:
3692:
3670:
3487:
3458:
3432:
3405:optical aberration
3390:
3331:
3283:
3256:
3227:
3210:
3183:
3163:
3150:
3131:
3104:
3061:
3026:
3014:
2989:
2979:
2945:
2887:
2860:
2831:
2792:
2753:
2700:
2668:
2585:
2549:
2525:
2436:
2401:
2374:
2345:
2313:
2281:
2218:
2186:
2106:
2060:
1921:
1831:numerical aperture
1819:
1790:
1678:
1632:
1558:
1488:
1416:
1396:
1376:
1331:
1304:
1275:
1090:
1078:
1062:
1050:
987:
943:
923:
903:
860:Focused laser beam
834:
802:
757:
730:
680:
660:
640:
584:
543:Rayleigh criterion
529:
497:Rayleigh criterion
493:angular resolution
481:
459:
435:
395:
338:
316:
296:
274:
231:
69:
57:
51:beam through a 90-
41:
29:
5885:Oldham Optical UK
5772:(10): 2120–2130.
5608:(10): 1819–1829.
5413:Giancoli, D. C.,
5381:10.1119/1.3204117
5347:Sidgwick, J. B.,
5219:Amateur astronomy
5194:
5087:
4924:{\displaystyle b}
4904:{\displaystyle a}
4851:
4620:
4605:
4523:{\displaystyle a}
4456:{\displaystyle f}
4416:{\displaystyle D}
4173:
3951:
3891:{\displaystyle R}
3867:{\displaystyle N}
3827:{\displaystyle R}
3802:
3657:
3617:
3580:
3386:
3327:
3186:{\displaystyle q}
3127:
3119:
2993:
2992:
2523:
2479:
2184:
2055:
2054:
2024:
2023:
1989:
1919:
1898:
1785:
1769:
1748:
1722:
1483:
1473:
1399:{\displaystyle a}
1263:
1204:
1106:Fourier transform
926:{\displaystyle a}
906:{\displaystyle R}
805:{\displaystyle x}
755:
725:
683:{\displaystyle f}
663:{\displaystyle x}
635:
618:
579:
430:
390:
269:
229:
5940:
5914:
5906:
5905:
5872:
5856:
5854:
5852:
5835:
5833:
5831:
5818:
5798:
5797:
5761:
5755:
5752:
5746:
5745:
5743:
5741:
5726:
5720:
5719:
5691:
5685:
5684:
5648:
5642:
5641:
5593:
5587:
5586:
5568:
5548:
5542:
5533:
5527:
5520:
5514:
5507:
5501:
5500:
5498:
5497:
5487:
5481:
5480:
5456:
5450:
5448:
5433:(2nd ed.).
5426:
5420:
5411:
5405:
5395:
5389:
5388:
5383:. Archived from
5360:
5354:
5345:
5339:
5329:
5323:
5322:
5298:
5292:
5291:
5279:
5208:
5206:
5205:
5200:
5195:
5193:
5192:
5180:
5179:
5164:
5163:
5151:
5150:
5138:
5117:
5115:
5114:
5109:
5098:
5097:
5088:
5083:
5075:
5052:
5051:
5047:
5046:
5029:
5028:
5024:
5023:
5006:
5005:
4998:
4997:
4988:
4983:
4982:
4970:
4969:
4960:
4955:
4954:
4930:
4928:
4927:
4922:
4910:
4908:
4907:
4902:
4887:
4885:
4884:
4879:
4862:
4861:
4852:
4844:
4821:
4820:
4801:
4796:
4744:
4739:
4720:
4715:
4700:
4699:
4668:
4660:
4641:
4636:
4624:
4623:
4622:
4621:
4613:
4607:
4606:
4598:
4580:
4572:
4553:
4548:
4529:
4527:
4526:
4521:
4501:
4499:
4498:
4493:
4491:
4490:
4481:
4462:
4460:
4459:
4454:
4442:
4440:
4439:
4434:
4422:
4420:
4419:
4414:
4402:
4400:
4399:
4394:
4389:
4384:
4383:
4358:
4356:
4355:
4350:
4348:
4347:
4331:
4329:
4328:
4323:
4318:
4317:
4308:
4307:
4295:
4284:
4283:
4268:
4267:
4224:
4222:
4221:
4216:
4196:
4194:
4193:
4188:
4186:
4182:
4174:
4169:
4156:
4155:
4137:
4136:
4126:
4123:
4118:
4100:
4096:
4082:
4077:
4052:
4047:
4027:
4026:
4004:
3999:
3977:
3972:
3952:
3950:
3946:
3945:
3923:
3897:
3895:
3894:
3889:
3873:
3871:
3870:
3865:
3853:
3851:
3850:
3845:
3833:
3831:
3830:
3825:
3813:
3811:
3810:
3805:
3803:
3801:
3793:
3785:
3746:
3744:
3743:
3738:
3733:
3729:
3701:
3699:
3698:
3693:
3679:
3677:
3676:
3671:
3669:
3668:
3663:
3659:
3658:
3653:
3640:
3639:
3623:
3618:
3613:
3603:
3602:
3589:
3581:
3579:
3578:
3577:
3568:
3567:
3548:
3547:
3538:
3496:
3494:
3493:
3488:
3467:
3465:
3464:
3459:
3441:
3439:
3438:
3433:
3431:
3430:
3399:
3397:
3396:
3391:
3384:
3371:
3370:
3340:
3338:
3337:
3332:
3325:
3312:
3311:
3292:
3290:
3289:
3284:
3282:
3281:
3265:
3263:
3262:
3257:
3252:
3251:
3235:
3219:
3217:
3216:
3211:
3209:
3208:
3192:
3190:
3189:
3184:
3172:
3170:
3169:
3164:
3158:
3140:
3138:
3137:
3132:
3125:
3124:
3120:
3117:
3112:
3103:
3102:
3101:
3085:
3069:
3031:root mean square
3023:
3021:
3020:
3015:
2961:
2960:
2954:
2952:
2951:
2946:
2917:
2916:
2899:Bessel functions
2896:
2894:
2893:
2888:
2886:
2885:
2869:
2867:
2866:
2861:
2859:
2858:
2840:
2838:
2837:
2832:
2805:
2800:
2766:
2761:
2743:
2742:
2709:
2707:
2706:
2701:
2677:
2675:
2674:
2669:
2661:
2660:
2651:
2650:
2638:
2627:
2626:
2611:
2610:
2594:
2592:
2591:
2586:
2584:
2583:
2558:
2556:
2555:
2550:
2548:
2534:
2532:
2531:
2526:
2524:
2522:
2521:
2520:
2511:
2510:
2500:
2496:
2495:
2485:
2480:
2478:
2477:
2476:
2463:
2462:
2461:
2451:
2446:
2441:
2434:
2429:
2428:
2410:
2408:
2407:
2402:
2400:
2399:
2383:
2381:
2380:
2375:
2373:
2372:
2354:
2352:
2351:
2346:
2322:
2320:
2319:
2314:
2290:
2288:
2287:
2282:
2280:
2275:
2261:
2247:
2246:
2227:
2225:
2224:
2219:
2195:
2193:
2192:
2187:
2185:
2180:
2178:
2164:
2150:
2149:
2115:
2113:
2112:
2107:
2096:
2069:
2067:
2066:
2061:
2056:
2047:
2046:
2034:
2030:
2025:
2022:
2021:
2009:
2008:
1999:
1995:
1990:
1988:
1977:
1930:
1928:
1927:
1922:
1920:
1918:
1907:
1899:
1891:
1889:
1878:
1877:
1856:
1855:
1828:
1826:
1825:
1820:
1818:
1817:
1799:
1797:
1796:
1791:
1786:
1778:
1770:
1768:
1757:
1749:
1747:
1736:
1728:
1723:
1721:
1710:
1687:
1685:
1684:
1679:
1665:
1641:
1639:
1638:
1633:
1567:
1565:
1564:
1559:
1548:
1547:
1497:
1495:
1494:
1489:
1484:
1476:
1474:
1469:
1458:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1375:
1370:
1365:
1340:
1338:
1337:
1332:
1330:
1329:
1313:
1311:
1310:
1305:
1303:
1302:
1284:
1282:
1281:
1276:
1274:
1273:
1268:
1264:
1259:
1249:
1248:
1235:
1228:
1227:
1215:
1214:
1209:
1205:
1203:
1185:
1162:
1161:
1148:
1141:
1140:
1087:
1085:
1084:
1079:
1059:
1057:
1056:
1051:
1046:
1041:
1040:
996:
994:
993:
988:
980:
975:
974:
952:
950:
949:
944:
932:
930:
929:
924:
912:
910:
909:
904:
818:image resolution
811:
809:
808:
803:
778:
776:
766:
764:
763:
758:
756:
748:
739:
737:
736:
731:
726:
721:
712:
689:
687:
686:
681:
669:
667:
666:
661:
649:
647:
646:
641:
636:
628:
619:
611:
593:
591:
590:
585:
580:
572:
490:
488:
487:
482:
480:
468:
466:
465:
460:
458:
444:
442:
441:
436:
431:
423:
404:
402:
401:
396:
391:
383:
375:
374:
373:
347:
345:
344:
339:
337:
325:
323:
322:
317:
305:
303:
302:
297:
283:
281:
280:
275:
270:
262:
240:
238:
237:
232:
230:
222:
102:with a circular
5948:
5947:
5943:
5942:
5941:
5939:
5938:
5937:
5928:Physical optics
5918:
5917:
5909:
5850:
5848:
5829:
5827:
5806:
5801:
5762:
5758:
5753:
5749:
5739:
5737:
5727:
5723:
5692:
5688:
5649:
5645:
5594:
5590:
5549:
5545:
5534:
5530:
5521:
5517:
5508:
5504:
5495:
5493:
5489:
5488:
5484:
5477:
5457:
5453:
5445:
5427:
5423:
5412:
5408:
5404:, 1835, p. 288.
5396:
5392:
5361:
5357:
5346:
5342:
5338:, 1835, p. 287.
5330:
5326:
5299:
5295:
5280:
5276:
5272:
5264:Speckle pattern
5215:
5188:
5184:
5175:
5171:
5159:
5155:
5146:
5142:
5137:
5129:
5126:
5125:
5093:
5089:
5076:
5074:
5042:
5038:
5034:
5030:
5019:
5015:
5011:
5007:
4993:
4989:
4984:
4978:
4974:
4965:
4961:
4956:
4950:
4946:
4945:
4941:
4939:
4936:
4935:
4916:
4913:
4912:
4896:
4893:
4892:
4857:
4853:
4843:
4816:
4812:
4797:
4792:
4740:
4735:
4716:
4711:
4680:
4676:
4661:
4656:
4637:
4632:
4612:
4611:
4597:
4596:
4592:
4588:
4573:
4568:
4549:
4544:
4538:
4535:
4534:
4515:
4512:
4511:
4508:
4486:
4482:
4477:
4472:
4469:
4468:
4448:
4445:
4444:
4428:
4425:
4424:
4408:
4405:
4404:
4385:
4379:
4375:
4364:
4361:
4360:
4343:
4339:
4337:
4334:
4333:
4313:
4309:
4303:
4299:
4291:
4279:
4275:
4248:
4244:
4242:
4239:
4238:
4235:
4204:
4201:
4200:
4151:
4147:
4132:
4128:
4127:
4125:
4119:
4114:
4078:
4073:
4048:
4043:
4032:
4028:
4022:
4018:
4000:
3995:
3973:
3968:
3957:
3953:
3941:
3937:
3927:
3922:
3905:
3902:
3901:
3883:
3880:
3879:
3859:
3856:
3855:
3839:
3836:
3835:
3819:
3816:
3815:
3794:
3786:
3784:
3752:
3749:
3748:
3713:
3709:
3707:
3704:
3703:
3687:
3684:
3683:
3664:
3635:
3631:
3624:
3622:
3598:
3594:
3590:
3588:
3587:
3583:
3582:
3573:
3569:
3563:
3559:
3549:
3543:
3539:
3537:
3520:
3517:
3516:
3502:
3473:
3470:
3469:
3447:
3444:
3443:
3426:
3422:
3420:
3417:
3416:
3366:
3362:
3360:
3357:
3356:
3307:
3303:
3301:
3298:
3297:
3277:
3273:
3271:
3268:
3267:
3247:
3243:
3231:
3225:
3222:
3221:
3204:
3200:
3198:
3195:
3194:
3178:
3175:
3174:
3154:
3148:
3145:
3144:
3113:
3108:
3097:
3093:
3086:
3084:
3080:
3065:
3044:
3041:
3040:
3009:
3006:
3005:
2998:
2912:
2908:
2906:
2903:
2902:
2881:
2877:
2875:
2872:
2871:
2854:
2850:
2848:
2845:
2844:
2801:
2796:
2762:
2757:
2738:
2734:
2717:
2714:
2713:
2686:
2683:
2682:
2656:
2652:
2646:
2642:
2634:
2622:
2618:
2606:
2602:
2600:
2597:
2596:
2579:
2575:
2564:
2561:
2560:
2544:
2542:
2539:
2538:
2516:
2512:
2506:
2502:
2501:
2491:
2487:
2486:
2484:
2472:
2468:
2464:
2457:
2453:
2447:
2442:
2437:
2435:
2433:
2424:
2420:
2418:
2415:
2414:
2395:
2391:
2389:
2386:
2385:
2368:
2364:
2362:
2359:
2358:
2328:
2325:
2324:
2296:
2293:
2292:
2276:
2271:
2257:
2242:
2238:
2233:
2230:
2229:
2201:
2198:
2197:
2179:
2174:
2160:
2145:
2141:
2136:
2133:
2132:
2120:of a camera is
2092:
2081:
2078:
2077:
2042:
2038:
2029:
2017:
2013:
2004:
2000:
1994:
1981:
1976:
1968:
1965:
1964:
1911:
1906:
1890:
1885:
1873:
1869:
1851:
1847:
1845:
1842:
1841:
1813:
1809:
1807:
1804:
1803:
1777:
1761:
1756:
1737:
1729:
1727:
1714:
1709:
1695:
1692:
1691:
1661:
1647:
1644:
1643:
1573:
1570:
1569:
1543:
1539:
1537:
1534:
1533:
1475:
1459:
1457:
1431:
1428:
1427:
1411:
1408:
1407:
1391:
1388:
1387:
1371:
1366:
1358:
1350:
1347:
1346:
1343:Bessel function
1325:
1321:
1319:
1316:
1315:
1298:
1294:
1292:
1289:
1288:
1269:
1244:
1240:
1236:
1234:
1230:
1229:
1223:
1219:
1210:
1186:
1157:
1153:
1149:
1147:
1143:
1142:
1136:
1132:
1115:
1112:
1111:
1102:squared modulus
1073:
1070:
1069:
1042:
1036:
1032:
1024:
1021:
1020:
1013:
976:
970:
966:
958:
955:
954:
938:
935:
934:
918:
915:
914:
898:
895:
894:
883:
871:
862:
826:
797:
794:
793:
774:
773:
747:
745:
742:
741:
713:
711:
699:
696:
695:
675:
672:
671:
655:
652:
651:
627:
610:
608:
605:
604:
571:
553:
550:
549:
534:
520:
476:
474:
471:
470:
454:
452:
449:
448:
422:
414:
411:
410:
382:
360:
359:
355:
353:
350:
349:
333:
331:
328:
327:
311:
308:
307:
306:is in radians,
291:
288:
287:
261:
250:
247:
246:
221:
204:
201:
200:
195:
145:seen through a
98:that a perfect
17:
12:
11:
5:
5946:
5936:
5935:
5930:
5916:
5915:
5907:
5888:
5878:
5864:
5863:
5862:
5841:
5805:
5804:External links
5802:
5800:
5799:
5756:
5747:
5721:
5686:
5653:Applied Optics
5643:
5602:Applied Optics
5588:
5543:
5528:
5515:
5502:
5482:
5475:
5451:
5443:
5435:Addison Wesley
5421:
5406:
5390:
5387:on 2011-05-14.
5375:(6): 362–365.
5355:
5340:
5324:
5293:
5273:
5271:
5268:
5267:
5266:
5261:
5256:
5251:
5246:
5241:
5239:Newton's rings
5236:
5231:
5226:
5221:
5214:
5211:
5210:
5209:
5198:
5191:
5187:
5183:
5178:
5174:
5170:
5167:
5162:
5158:
5154:
5149:
5145:
5141:
5136:
5133:
5119:
5118:
5107:
5104:
5101:
5096:
5092:
5086:
5082:
5079:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5050:
5045:
5041:
5037:
5033:
5027:
5022:
5018:
5014:
5010:
5004:
5001:
4996:
4992:
4987:
4981:
4977:
4973:
4968:
4964:
4959:
4953:
4949:
4944:
4920:
4900:
4889:
4888:
4877:
4874:
4871:
4868:
4865:
4860:
4856:
4850:
4847:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4819:
4815:
4811:
4808:
4805:
4800:
4795:
4791:
4787:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4743:
4738:
4734:
4730:
4727:
4724:
4719:
4714:
4710:
4706:
4703:
4698:
4695:
4692:
4689:
4686:
4683:
4679:
4675:
4672:
4667:
4664:
4659:
4655:
4651:
4648:
4645:
4640:
4635:
4631:
4627:
4619:
4616:
4610:
4604:
4601:
4595:
4591:
4587:
4584:
4579:
4576:
4571:
4567:
4563:
4560:
4557:
4552:
4547:
4543:
4519:
4507:
4504:
4489:
4485:
4480:
4476:
4452:
4432:
4412:
4392:
4388:
4382:
4378:
4374:
4371:
4368:
4346:
4342:
4321:
4316:
4312:
4306:
4302:
4298:
4294:
4290:
4287:
4282:
4278:
4274:
4271:
4266:
4263:
4260:
4257:
4254:
4251:
4247:
4234:
4231:
4214:
4211:
4208:
4185:
4181:
4178:
4172:
4168:
4165:
4162:
4159:
4154:
4150:
4146:
4143:
4140:
4135:
4131:
4122:
4117:
4113:
4109:
4106:
4103:
4099:
4095:
4092:
4089:
4086:
4081:
4076:
4072:
4068:
4065:
4062:
4059:
4056:
4051:
4046:
4042:
4038:
4035:
4031:
4025:
4021:
4017:
4014:
4011:
4008:
4003:
3998:
3994:
3990:
3987:
3984:
3981:
3976:
3971:
3967:
3963:
3960:
3956:
3949:
3944:
3940:
3936:
3933:
3930:
3926:
3921:
3918:
3915:
3912:
3909:
3887:
3863:
3843:
3823:
3800:
3797:
3792:
3789:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3736:
3732:
3728:
3725:
3722:
3719:
3716:
3712:
3691:
3667:
3662:
3656:
3652:
3649:
3646:
3643:
3638:
3634:
3630:
3627:
3621:
3616:
3612:
3609:
3606:
3601:
3597:
3593:
3586:
3576:
3572:
3566:
3562:
3558:
3555:
3552:
3546:
3542:
3536:
3533:
3530:
3527:
3524:
3501:
3498:
3486:
3483:
3480:
3477:
3468:as opposed to
3457:
3454:
3451:
3429:
3425:
3401:
3400:
3389:
3383:
3380:
3377:
3374:
3369:
3365:
3342:
3341:
3330:
3324:
3321:
3318:
3315:
3310:
3306:
3280:
3276:
3255:
3250:
3246:
3242:
3238:
3234:
3230:
3207:
3203:
3182:
3161:
3157:
3153:
3130:
3123:
3116:
3111:
3107:
3100:
3096:
3092:
3089:
3083:
3079:
3076:
3072:
3068:
3064:
3060:
3057:
3054:
3051:
3048:
3013:
2997:
2994:
2991:
2990:
2980:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2915:
2911:
2884:
2880:
2857:
2853:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2804:
2799:
2795:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2765:
2760:
2756:
2752:
2749:
2746:
2741:
2737:
2733:
2730:
2727:
2724:
2721:
2699:
2696:
2693:
2690:
2667:
2664:
2659:
2655:
2649:
2645:
2641:
2637:
2633:
2630:
2625:
2621:
2617:
2614:
2609:
2605:
2582:
2578:
2574:
2571:
2568:
2547:
2519:
2515:
2509:
2505:
2499:
2494:
2490:
2483:
2475:
2471:
2467:
2460:
2456:
2450:
2445:
2440:
2432:
2427:
2423:
2398:
2394:
2371:
2367:
2357:The intensity
2344:
2341:
2338:
2335:
2332:
2312:
2309:
2306:
2303:
2300:
2279:
2274:
2270:
2267:
2264:
2260:
2256:
2253:
2250:
2245:
2241:
2237:
2217:
2214:
2211:
2208:
2205:
2183:
2177:
2173:
2170:
2167:
2163:
2159:
2156:
2153:
2148:
2144:
2140:
2105:
2102:
2099:
2095:
2091:
2088:
2085:
2059:
2053:
2050:
2045:
2041:
2037:
2033:
2028:
2020:
2016:
2012:
2007:
2003:
1998:
1993:
1987:
1984:
1980:
1975:
1972:
1917:
1914:
1910:
1905:
1902:
1897:
1894:
1888:
1884:
1881:
1876:
1872:
1868:
1865:
1862:
1859:
1854:
1850:
1816:
1812:
1789:
1784:
1781:
1776:
1773:
1767:
1764:
1760:
1755:
1752:
1746:
1743:
1740:
1735:
1732:
1726:
1720:
1717:
1713:
1708:
1705:
1702:
1699:
1677:
1674:
1671:
1668:
1664:
1660:
1657:
1654:
1651:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1557:
1554:
1551:
1546:
1542:
1487:
1482:
1479:
1472:
1468:
1465:
1462:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1415:
1395:
1374:
1369:
1364:
1361:
1357:
1354:
1328:
1324:
1301:
1297:
1272:
1267:
1262:
1258:
1255:
1252:
1247:
1243:
1239:
1233:
1226:
1222:
1218:
1213:
1208:
1202:
1199:
1196:
1193:
1189:
1184:
1181:
1178:
1175:
1172:
1168:
1165:
1160:
1156:
1152:
1146:
1139:
1135:
1131:
1128:
1125:
1122:
1119:
1077:
1049:
1045:
1039:
1035:
1031:
1028:
1012:
1009:
986:
983:
979:
973:
969:
965:
962:
942:
922:
902:
882:
879:
870:
867:
861:
858:
825:
822:
801:
754:
751:
729:
724:
720:
716:
709:
706:
703:
679:
659:
639:
634:
631:
625:
622:
617:
614:
583:
578:
575:
569:
566:
563:
560:
557:
533:
530:
519:
516:
479:
457:
434:
429:
426:
421:
418:
394:
389:
386:
381:
378:
372:
369:
366:
363:
358:
336:
315:
295:
273:
268:
265:
260:
257:
254:
228:
225:
220:
217:
214:
211:
208:
194:
191:
127:central region
15:
9:
6:
4:
3:
2:
5945:
5934:
5931:
5929:
5926:
5925:
5923:
5912:
5908:
5903:
5902:
5897:
5894:
5889:
5886:
5882:
5879:
5876:
5870:
5867:Paul Padley.
5865:
5860:
5847:
5842:
5839:
5826:
5821:
5820:
5816:
5812:
5808:
5807:
5795:
5791:
5787:
5783:
5779:
5775:
5771:
5767:
5760:
5751:
5736:
5732:
5725:
5717:
5713:
5709:
5705:
5701:
5697:
5690:
5682:
5678:
5674:
5670:
5666:
5662:
5658:
5654:
5647:
5639:
5635:
5631:
5627:
5623:
5619:
5615:
5611:
5607:
5603:
5599:
5592:
5584:
5580:
5576:
5572:
5567:
5562:
5558:
5554:
5547:
5540:
5539:
5532:
5525:
5519:
5512:
5506:
5492:
5486:
5478:
5476:0-07-134916-2
5472:
5468:
5464:
5463:
5455:
5446:
5444:0-201-11609-X
5440:
5436:
5432:
5425:
5418:
5417:
5410:
5403:
5401:
5394:
5386:
5382:
5378:
5374:
5370:
5366:
5359:
5352:
5351:
5344:
5337:
5335:
5328:
5320:
5316:
5312:
5308:
5304:
5297:
5289:
5285:
5278:
5274:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5242:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5216:
5196:
5189:
5181:
5176:
5172:
5165:
5160:
5152:
5147:
5143:
5134:
5131:
5124:
5123:
5122:
5102:
5094:
5090:
5084:
5080:
5077:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5048:
5043:
5039:
5035:
5031:
5025:
5020:
5016:
5012:
5008:
5002:
4999:
4994:
4990:
4985:
4979:
4975:
4971:
4966:
4962:
4957:
4951:
4947:
4942:
4934:
4933:
4932:
4918:
4898:
4875:
4869:
4866:
4858:
4854:
4848:
4845:
4840:
4837:
4834:
4828:
4825:
4817:
4813:
4809:
4806:
4803:
4798:
4793:
4789:
4785:
4782:
4779:
4773:
4770:
4767:
4764:
4761:
4755:
4752:
4749:
4746:
4741:
4736:
4732:
4728:
4725:
4722:
4717:
4712:
4708:
4704:
4701:
4696:
4693:
4690:
4687:
4684:
4681:
4677:
4673:
4670:
4665:
4662:
4657:
4653:
4649:
4646:
4643:
4638:
4633:
4629:
4625:
4614:
4608:
4599:
4593:
4589:
4585:
4582:
4577:
4574:
4569:
4565:
4561:
4558:
4555:
4550:
4545:
4541:
4533:
4532:
4531:
4517:
4503:
4487:
4483:
4478:
4474:
4464:
4450:
4430:
4410:
4390:
4386:
4380:
4376:
4372:
4369:
4366:
4344:
4340:
4314:
4310:
4304:
4300:
4292:
4285:
4280:
4276:
4269:
4264:
4261:
4258:
4255:
4252:
4249:
4245:
4230:
4226:
4212:
4206:
4197:
4183:
4179:
4176:
4170:
4163:
4160:
4152:
4148:
4141:
4133:
4129:
4120:
4115:
4111:
4107:
4104:
4101:
4097:
4090:
4087:
4079:
4074:
4070:
4066:
4060:
4057:
4049:
4044:
4040:
4036:
4033:
4029:
4023:
4019:
4015:
4009:
4001:
3996:
3992:
3988:
3982:
3974:
3969:
3965:
3961:
3958:
3954:
3942:
3938:
3934:
3931:
3924:
3919:
3913:
3907:
3899:
3885:
3877:
3861:
3841:
3821:
3798:
3795:
3790:
3787:
3781:
3775:
3769:
3766:
3763:
3760:
3757:
3754:
3734:
3730:
3726:
3723:
3720:
3717:
3714:
3710:
3689:
3680:
3665:
3660:
3654:
3647:
3644:
3636:
3632:
3628:
3625:
3619:
3614:
3607:
3599:
3595:
3591:
3584:
3574:
3564:
3560:
3556:
3553:
3544:
3540:
3534:
3528:
3522:
3514:
3512:
3508:
3497:
3484:
3481:
3478:
3475:
3455:
3452:
3449:
3427:
3423:
3414:
3410:
3406:
3387:
3381:
3378:
3375:
3372:
3367:
3363:
3355:
3354:
3353:
3351:
3347:
3328:
3322:
3319:
3316:
3313:
3308:
3304:
3296:
3295:
3294:
3278:
3274:
3253:
3248:
3244:
3240:
3236:
3232:
3228:
3205:
3201:
3180:
3159:
3155:
3151:
3141:
3128:
3121:
3114:
3109:
3105:
3098:
3094:
3090:
3087:
3081:
3077:
3074:
3070:
3066:
3062:
3058:
3052:
3046:
3038:
3036:
3032:
3011:
3002:
2985:
2981:
2977: =
2976:
2972:
2967:
2963:
2962:
2959:
2956:
2942:
2939:
2933:
2930:
2927:
2924:
2921:
2913:
2909:
2900:
2882:
2878:
2855:
2851:
2841:
2822:
2819:
2816:
2813:
2810:
2802:
2797:
2793:
2789:
2783:
2780:
2777:
2774:
2771:
2763:
2758:
2754:
2750:
2747:
2739:
2735:
2731:
2725:
2719:
2711:
2694:
2688:
2679:
2665:
2657:
2653:
2647:
2643:
2635:
2628:
2623:
2619:
2612:
2607:
2603:
2580:
2576:
2572:
2569:
2566:
2535:
2517:
2513:
2507:
2503:
2497:
2492:
2488:
2481:
2473:
2469:
2465:
2458:
2454:
2448:
2443:
2430:
2425:
2421:
2412:
2396:
2392:
2369:
2365:
2355:
2342:
2339:
2336:
2333:
2330:
2310:
2307:
2304:
2301:
2298:
2277:
2272:
2268:
2265:
2262:
2258:
2251:
2243:
2239:
2235:
2215:
2212:
2209:
2206:
2203:
2181:
2175:
2171:
2168:
2165:
2161:
2154:
2146:
2142:
2138:
2129:
2127:
2123:
2119:
2103:
2100:
2097:
2093:
2089:
2086:
2083:
2075:
2070:
2057:
2051:
2048:
2043:
2039:
2035:
2031:
2026:
2018:
2014:
2010:
2005:
2001:
1996:
1991:
1985:
1982:
1978:
1973:
1970:
1962:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1931:
1915:
1912:
1908:
1903:
1900:
1895:
1892:
1886:
1882:
1879:
1874:
1870:
1866:
1863:
1860:
1857:
1852:
1848:
1839:
1835:
1832:
1814:
1810:
1800:
1787:
1782:
1779:
1774:
1771:
1765:
1762:
1758:
1753:
1750:
1744:
1741:
1738:
1733:
1730:
1724:
1718:
1715:
1711:
1706:
1703:
1700:
1697:
1689:
1675:
1672:
1669:
1666:
1662:
1658:
1655:
1652:
1649:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1552:
1544:
1540:
1532:The zeros of
1530:
1528:
1524:
1519:
1517:
1513:
1509:
1505:
1501:
1485:
1480:
1477:
1470:
1466:
1463:
1460:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1413:
1393:
1372:
1367:
1362:
1359:
1355:
1352:
1344:
1326:
1322:
1299:
1295:
1285:
1270:
1265:
1260:
1253:
1245:
1241:
1237:
1231:
1224:
1220:
1216:
1211:
1206:
1200:
1197:
1194:
1191:
1187:
1179:
1176:
1173:
1170:
1166:
1158:
1154:
1150:
1144:
1137:
1133:
1129:
1123:
1117:
1109:
1107:
1103:
1099:
1095:
1075:
1066:
1047:
1043:
1037:
1033:
1029:
1026:
1017:
1008:
1004:
1001:
997:
984:
981:
977:
971:
967:
963:
960:
940:
920:
900:
890:
888:
878:
876:
866:
857:
855:
850:
846:
842:
839:
830:
824:The human eye
821:
819:
815:
799:
791:
787:
783:
782:Sunny 16 rule
779:
770:
752:
749:
727:
722:
718:
714:
707:
704:
701:
693:
677:
657:
637:
632:
629:
623:
620:
615:
612:
601:
599:
594:
581:
576:
573:
567:
564:
561:
558:
555:
546:
544:
540:
539:just resolved
524:
515:
510:
505:
501:
498:
494:
477:
455:
445:
432:
427:
424:
419:
416:
408:
405:
392:
387:
384:
379:
376:
356:
334:
313:
293:
284:
271:
266:
263:
258:
255:
252:
244:
241:
226:
223:
218:
215:
212:
209:
206:
198:
190:
188:
184:
180:
176:
171:
169:
164:
160:
156:
154:
153:
148:
144:
140:
139:John Herschel
136:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
93:
90:
86:
82:
78:
74:
66:
61:
54:
50:
45:
38:
33:
26:
21:
5899:
5884:
5874:
5858:
5849:. Retrieved
5837:
5828:. Retrieved
5769:
5765:
5759:
5750:
5738:. Retrieved
5734:
5724:
5699:
5695:
5689:
5659:(14): 2404.
5656:
5652:
5646:
5605:
5601:
5591:
5556:
5552:
5546:
5536:
5531:
5523:
5518:
5505:
5494:. Retrieved
5485:
5461:
5454:
5430:
5424:
5415:
5409:
5399:
5393:
5385:the original
5372:
5368:
5358:
5349:
5343:
5333:
5327:
5310:
5306:
5296:
5287:
5277:
5259:Strehl ratio
5244:Optical unit
5120:
4890:
4509:
4465:
4236:
4227:
4198:
3900:
3681:
3515:
3503:
3408:
3402:
3345:
3343:
3142:
3039:
3027:
2974:
2970:
2957:
2842:
2712:
2680:
2536:
2413:
2356:
2291:) occurs at
2196:) occurs at
2130:
2073:
2071:
1963:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1932:
1833:
1801:
1690:
1531:
1522:
1520:
1507:
1503:
1499:
1286:
1110:
1091:
1005:
1002:
998:
891:
884:
872:
869:Aiming sight
863:
835:
814:image sensor
692:focal length
602:
597:
595:
547:
538:
535:
512:
507:
502:
446:
409:
406:
285:
245:
242:
199:
196:
172:
167:
165:
162:
158:
150:
124:
85:Airy pattern
84:
80:
76:
70:
37:D65 spectrum
5933:Diffraction
5449:Sect. 5.7.1
5224:Apodization
2126:diffraction
179:microscopes
108:diffraction
65:camera lens
5922:Categories
5875:Connexions
5840:(website).
5702:(4): 470.
5566:2309.09827
5559:: 168679.
5522:E. Hecht,
5496:2023-12-18
5313:: 283–91.
2337:5.13562230
2305:2.58383899
2210:1.61633995
2118:resolution
1527:collimated
790:cone cells
786:nanometers
183:telescopes
168:appearance
131:concentric
53:micrometre
5901:MathWorld
5630:2155-3165
5135:≡
5072:π
5000:≤
4943:∫
4841:π
4790:∫
4786:π
4774:ϕ
4771:
4756:
4750:ϕ
4742:π
4733:∫
4709:∫
4697:ϕ
4694:
4674:ϕ
4666:π
4654:∫
4630:∫
4618:→
4609:⋅
4603:→
4586:ϕ
4578:π
4566:∫
4542:∫
4431:λ
4373:π
4301:λ
4229:mirrors.
4210:→
4207:ϵ
4161:ϵ
4112:∫
4108:ϵ
4102:−
4088:ϵ
4067:−
4058:ϵ
4037:−
4020:ϵ
3989:−
3962:−
3939:ϵ
3935:−
3842:λ
3796:λ
3788:π
3782:≈
3776:θ
3770:
3721:ϵ
3718:≤
3690:ϵ
3645:ϵ
3629:ϵ
3620:−
3561:ϵ
3557:−
3479:λ
3453:λ
3424:ω
3379:λ
3373:≈
3364:ω
3320:λ
3314:≈
3305:ω
3275:ω
3202:ω
3106:ω
3088:−
3078:
3059:≈
3012:λ
2934:θ
2931:
2823:θ
2820:
2790:−
2784:θ
2781:
2751:−
2726:θ
2695:θ
2644:λ
2573:π
2504:λ
2340:…
2308:…
2213:…
2087:≈
1909:λ
1893:λ
1880:≈
1871:θ
1867:
1780:λ
1759:λ
1742:π
1734:λ
1707:≈
1704:θ
1701:
1673:…
1663:θ
1659:
1627:…
1597:≈
1594:θ
1591:
1471:λ
1464:π
1452:θ
1449:
1414:θ
1373:λ
1363:π
1201:θ
1198:
1180:θ
1177:
1124:θ
1094:intensity
1076:θ
1048:λ
1030:≫
982:λ
941:λ
845:human eye
715:λ
630:λ
574:λ
562:θ
559:
385:λ
357:θ
314:λ
294:θ
264:λ
256:≈
253:θ
224:λ
216:≈
213:θ
210:
147:telescope
120:astronomy
81:Airy disc
77:Airy disk
25:grayscale
5851:June 15,
5830:June 15,
5794:25401234
5681:18231508
5638:17356626
5402:, Vol. 5
5336:, Vol. 5
5213:See also
3876:f-number
3350:f-number
3237:′
3160:′
3071:′
3035:Gaussian
1986:′
1838:f-number
1510:, where
843:for the
841:f-number
769:f-number
518:Examples
104:aperture
5774:Bibcode
5740:May 18,
5704:Bibcode
5661:Bibcode
5610:Bibcode
5571:Bibcode
5315:Bibcode
5284:"Light"
3874:is the
3348:is the
2122:limited
1624:16.4706
1618:13.3237
1612:10.1735
1568:are at
1341:is the
1104:of the
838:fastest
767:is the
532:Cameras
175:cameras
112:physics
89:focused
5792:
5679:
5636:
5628:
5524:Optics
5473:
5441:
5431:Optics
5121:where
4332:where
3814:where
3682:where
3385:
3344:where
3326:
3143:where
3126:
2843:where
2537:where
1959:N= f/d
1670:3.8317
1606:7.0156
1600:3.8317
1498:where
1287:where
875:FN FNC
650:where
447:where
286:where
118:, and
116:optics
83:) and
75:, the
73:optics
5561:arXiv
1840:) by
854:fovea
788:(see
780:(see
380:1.029
96:light
49:laser
5853:2006
5832:2006
5790:PMID
5742:2013
5677:PMID
5634:PMID
5626:ISSN
5557:1056
5509:See
5471:ISBN
5439:ISBN
4911:and
4199:For
3724:<
3509:and
3476:1.22
3450:0.84
3376:0.90
3317:0.84
2897:are
2870:and
2072:for
1943:edge
1904:1.22
1883:1.22
1775:1.22
1754:1.22
1731:3.83
1712:3.83
1523:does
1092:The
964:>
836:The
740:but
708:1.22
624:1.22
568:1.22
425:2.76
259:1.22
219:1.22
193:Size
181:and
143:star
100:lens
92:spot
79:(or
5782:doi
5712:doi
5669:doi
5618:doi
5579:doi
5377:doi
4768:cos
4753:cos
4691:cos
4530:is
3767:sin
3403:In
3075:exp
2973:sin
2928:sin
2817:sin
2778:sin
1864:sin
1698:sin
1688:or
1656:sin
1588:sin
1446:sin
1195:sin
1174:sin
556:sin
545:).
207:sin
94:of
71:In
5924::
5898:.
5883:,
5873:,
5813:.
5788:.
5780:.
5770:31
5768:.
5733:.
5710:.
5698:.
5675:.
5667:.
5657:25
5655:.
5632:.
5624:.
5616:.
5606:46
5604:.
5600:.
5577:.
5569:.
5555:.
5469:.
5465:.
5437:.
5373:47
5371:.
5367:.
5309:.
5305:.
5286:.
4931::
3513:.
2971:ka
2128:.
1518:.
777:/8
189:.
177:,
155::
122:.
114:,
5913:.
5904:.
5887:.
5871:.
5861:.
5855:.
5834:.
5817:.
5796:.
5784::
5776::
5744:.
5718:.
5714::
5706::
5700:3
5683:.
5671::
5663::
5640:.
5620::
5612::
5585:.
5581::
5573::
5563::
5499:.
5479:.
5447:.
5379::
5321:.
5317::
5311:5
5197:.
5190:2
5186:)
5182:b
5177:y
5173:k
5169:(
5166:+
5161:2
5157:)
5153:a
5148:x
5144:k
5140:(
5132:c
5106:)
5103:c
5100:(
5095:1
5091:J
5085:c
5081:b
5078:a
5069:2
5066:=
5063:y
5060:d
5057:x
5054:d
5049:y
5044:y
5040:k
5036:i
5032:e
5026:x
5021:x
5017:k
5013:i
5009:e
5003:1
4995:2
4991:b
4986:/
4980:2
4976:y
4972:+
4967:2
4963:a
4958:/
4952:2
4948:x
4919:b
4899:a
4876:.
4873:)
4870:r
4867:k
4864:(
4859:1
4855:J
4849:k
4846:a
4838:2
4835:=
4832:)
4829:r
4826:k
4823:(
4818:0
4814:J
4810:r
4807:d
4804:r
4799:a
4794:0
4783:2
4780:=
4777:)
4765:r
4762:k
4759:(
4747:d
4737:0
4729:r
4726:d
4723:r
4718:a
4713:0
4705:2
4702:=
4688:r
4685:k
4682:i
4678:e
4671:d
4663:2
4658:0
4650:r
4647:d
4644:r
4639:a
4634:0
4626:=
4615:r
4600:k
4594:i
4590:e
4583:d
4575:2
4570:0
4562:r
4559:d
4556:r
4551:a
4546:0
4518:a
4488:2
4484:e
4479:/
4475:1
4451:f
4411:D
4391:4
4387:/
4381:2
4377:D
4370:=
4367:A
4345:0
4341:P
4320:)
4315:2
4311:f
4305:2
4297:(
4293:/
4289:)
4286:A
4281:0
4277:P
4273:(
4270:=
4265:y
4262:r
4259:i
4256:A
4253:,
4250:0
4246:I
4213:0
4184:)
4180:t
4177:d
4171:t
4167:)
4164:t
4158:(
4153:1
4149:J
4145:)
4142:t
4139:(
4134:1
4130:J
4121:x
4116:0
4105:4
4098:]
4094:)
4091:x
4085:(
4080:2
4075:1
4071:J
4064:)
4061:x
4055:(
4050:2
4045:0
4041:J
4034:1
4030:[
4024:2
4016:+
4013:)
4010:x
4007:(
4002:2
3997:1
3993:J
3986:)
3983:x
3980:(
3975:2
3970:0
3966:J
3959:1
3955:(
3948:)
3943:2
3932:1
3929:(
3925:1
3920:=
3917:)
3914:R
3911:(
3908:E
3886:R
3862:N
3822:R
3799:N
3791:R
3779:)
3773:(
3764:a
3761:k
3758:=
3755:x
3735:,
3731:)
3727:1
3715:0
3711:(
3666:2
3661:)
3655:x
3651:)
3648:x
3642:(
3637:1
3633:J
3626:2
3615:x
3611:)
3608:x
3605:(
3600:1
3596:J
3592:2
3585:(
3575:2
3571:)
3565:2
3554:1
3551:(
3545:0
3541:I
3535:=
3532:)
3529:R
3526:(
3523:I
3485:.
3482:N
3456:N
3428:0
3388:.
3382:N
3368:0
3346:N
3329:,
3323:N
3309:0
3279:0
3254:,
3249:0
3245:I
3241:=
3233:0
3229:I
3206:0
3181:q
3156:0
3152:I
3129:,
3122:)
3115:2
3110:0
3099:2
3095:q
3091:2
3082:(
3067:0
3063:I
3056:)
3053:q
3050:(
3047:I
2975:θ
2943:0
2940:=
2937:)
2925:a
2922:k
2919:(
2914:1
2910:J
2883:1
2879:J
2856:0
2852:J
2829:]
2826:)
2814:a
2811:k
2808:(
2803:2
2798:1
2794:J
2787:)
2775:a
2772:k
2769:(
2764:2
2759:0
2755:J
2748:1
2745:[
2740:0
2736:P
2732:=
2729:)
2723:(
2720:P
2698:)
2692:(
2689:I
2666:.
2663:)
2658:2
2654:f
2648:2
2640:(
2636:/
2632:)
2629:A
2624:0
2620:P
2616:(
2613:=
2608:0
2604:I
2581:2
2577:a
2570:=
2567:A
2546:E
2518:2
2514:R
2508:2
2498:A
2493:0
2489:P
2482:=
2474:2
2470:R
2466:2
2459:2
2455:A
2449:2
2444:A
2439:E
2431:=
2426:0
2422:I
2397:0
2393:P
2370:0
2366:I
2343:.
2334:=
2331:x
2311:,
2302:=
2299:x
2278:e
2273:/
2269:1
2266:=
2263:x
2259:/
2255:)
2252:x
2249:(
2244:1
2240:J
2236:2
2216:;
2207:=
2204:x
2182:2
2176:/
2172:1
2169:=
2166:x
2162:/
2158:)
2155:x
2152:(
2147:1
2143:J
2139:2
2104:.
2101:N
2098:2
2094:/
2090:1
2084:A
2074:N
2058:;
2052:1
2049:+
2044:2
2040:N
2036:4
2032:1
2027:=
2019:2
2015:r
2011:+
2006:2
2002:f
1997:r
1992:=
1983:R
1979:r
1974:=
1971:A
1955:A
1951:f
1947:d
1939:d
1935:A
1916:A
1913:2
1901:=
1896:d
1887:R
1875:1
1861:R
1858:=
1853:1
1849:q
1834:A
1815:1
1811:q
1788:.
1783:d
1772:=
1766:a
1763:2
1751:=
1745:a
1739:2
1725:=
1719:a
1716:k
1676:,
1667:=
1653:a
1650:k
1630:.
1621:,
1615:,
1609:,
1603:,
1585:a
1582:k
1579:=
1576:x
1556:)
1553:x
1550:(
1545:1
1541:J
1508:R
1504:R
1500:q
1486:,
1481:R
1478:q
1467:a
1461:2
1455:=
1443:a
1440:k
1437:=
1434:x
1394:a
1368:/
1360:2
1356:=
1353:k
1327:1
1323:J
1300:0
1296:I
1271:2
1266:]
1261:x
1257:)
1254:x
1251:(
1246:1
1242:J
1238:2
1232:[
1225:0
1221:I
1217:=
1212:2
1207:]
1192:a
1188:k
1183:)
1171:a
1167:k
1164:(
1159:1
1155:J
1151:2
1145:[
1138:0
1134:I
1130:=
1127:)
1121:(
1118:I
1044:/
1038:2
1034:a
1027:R
985:.
978:/
972:2
968:a
961:R
921:a
901:R
800:x
775:f
753:d
750:f
728:,
723:d
719:f
705:=
702:x
678:f
658:x
638:,
633:d
621:=
616:f
613:x
598:d
582:.
577:d
565:=
478:a
456:s
433:,
428:a
420:=
417:s
393:.
388:d
377:=
371:M
368:H
365:W
362:F
335:d
272:,
267:d
227:d
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