Talk:Airy disk

Photoreceptor spacing

"The smallest f-number for the human eye is about 2.1. The resulting resolution is about 1 micro meter. This happens to be about the distance between optically sensitive cells, photoreceptors, in the human eye."

I dispute this, foveal cones are typically 1.5μm in diameter with a 0.5μm gap between each, thus giving the array a 2μm spacing. However this may not be the case for the entire retina so I will leave the article until someone more knowledgeable can confirm this —The preceding unsigned comment was added by 131.251.0.55 (talk) 22:29, 6 May 2007 (UTC).

I'm no eye expert, but it would make sense to have a 2µm spacing, as in visible light, directly applying the equation just before this statement, the airy disk is about 1µm, that's true, but for 400nm. The airy disk would then vary between 1µm and 1.8µm in the visible spectrum (taken here between 400 and 700nm). Then again, we would have to take into account the repartition of the three different cones... Palleas 14:37, 15 May 2007 (UTC)

The peak cone density of the human fovea is about 170,000 per square millimeter, which translates into a spacing of approximately 2.5 µm spacing, depending on how they are packed (reference included in the article). The section on the eye also neglected spherical aberration, which is the real limit on resolution at larger apertures. Visual acuity is a complicated subject that cannot be addressed accurately with these back-of-the-envelope calculations. See the article on the eye. This section of the article probably needs further cleanup.

--Drphysics (talk) 09:29, 20 September 2008 (UTC)

Spherical Aberration Image

Might this be better placed in the article on lens aberrations? neffk (talk) 15:12, 22 April 2008 (UTC)

Maybe request Airy beams

as per http://www.physorg.com/news140438326.html. Emesee (talk) 09:39, 13 September 2008 (UTC)

An Airy beam has little relation to Airy disks. It is more closely related to a Bessel beam. Airy beams could be included in a more general discussion of 'non-diffractive' light or unconventional solutions to Maxwell's equations along with Bessel beams and knotted light. See Light Beam with a Curve. --Drphysics (talk) 16:26, 20 September 2008 (UTC)

Page problems

The page has has problem generating formulas for the past couple of days. I'm just editing this page ni the hope that a change will alert someone to the need for service. Thanks!
65.202.227.47 (talk) 13:50, 1 October 2008 (UTC)mjd

Diffraction rings look too bright

Hi, I was wondering whether the right computation has been done to render the density plot at the top right corner of the article: the outer rings seem far too bright compared to the central peak. In other words, the rings should vanish much faster IMO. The right formula is to be of the (J₁(r)/r)² form, and the image showed here is more likely to represent something of the J₀(r)² form. PS: I've also made this statement on the page dedicated to the image. Cheers, MatP (talk) 16:26, 5 November 2008 (UTC)

It's some sort of logarithmic scale. The colours do match to the given legend, and it is noted to haved a "scaled" colour function - if you read the source code you can see exactly how it works. I was planning to update the image anyway, with a linear and logarithmic version.Inductiveload (talk) 08:43, 19 March 2009 (UTC)

lack of definition

The symbol d used in this page is not defined and must be guessed —Preceding unsigned comment added by 88.163.195.44 (talk) 08:13, 14 February 2010 (UTC)

Early in section 1 it includes:"where θ is in radians and λ is the wavelength of the light and d is the diameter of the aperture." Is that not sufficient?

Interferometrist (talk) 13:24, 24 April 2010 (UTC)

Math error: rms size of comparable gaussian beam

The section "Approximation using a Gaussian profile" is relevant for the exact reason stated: that the rms of the Airy function itself is infinite (but just in the mathematical sense). However I believe the math is wrong, because the rms of the gaussian IN TWO DIMENSIONS is not w but (according to a quick calculation) sqrt(2) w. I will change that but first need to find a verification in the public domain to reference. Tell me if you disagree.... Interferometrist (talk) 14:03, 24 April 2010 (UTC)

That sounds good. By the way, "RMS spot size" is a poorly defined term (and the wikilink to RMS is no help); a "spot size" is not an ensemble. ("RMS" means "root mean square". This is the (square) root of the mean of the squared value of what?) Geoffrey.landis (talk) 15:49, 21 January 2011 (UTC)

The ${\omega _{0}}$ in

$$I(q)\approx I'_{0}\exp \left({\frac {-2q^{2}}{\omega _{0}^{2}}}\right)\ ,$$

is the beam radius but not the ${\sigma }$ of the gaussian function but ${2\sigma }$ . So the RMS will be

${RMS=\sigma {\sqrt {2}}={\frac {\omega _{0}}{\sqrt {2}}}}$ 81.223.7.138 (talk) 14:46, 5 July 2023 (UTC)

I have edited this section to resolve inconsistencies in the equations and between the equations and plot. For now, I have parameterized the Gaussian in terms of its 1-dimensional rms width. However, Interferometrist is probably correct that the geometric spot size ought to be compared to the 2D rms size of the Gaussian. I would urge an expert on this topic to edit the section accordingly.

I don't know how you define in this case that RMS value. But the function $\left({\frac {J_{1}(x)}{x}}\right)^{2}$ is integrable, in fact $\int \left({\frac {J_{1}(x)}{x}}\right)^{2}dx={\frac {2}{3x}}\left[\left(-xJ_{2}(x)+{\frac {3}{2}}J_{1}(x)\right)^{2}+\left(x^{2}-{\frac {3}{4}}\right)J_{1}^{2}(x)\right]+c$ $\int \limits _{0}^{b}\left({\frac {J_{1}(x)}{x}}\right)^{2}dx={\frac {1}{3b}}\left[2b^{2}J_{0}^{2}(b)-2bJ_{0}(b)J_{1}(b)-J_{1}^{2}(b)+2J_{1}(b^{2})b^{2}\right]$

$\int \limits _{0}^{b}\left({\frac {J_{1}(x)}{x}}\right)^{2}dx={\frac {4}{3\pi }}\approx 0.4244131814$

Actually I verified the improper integral using numerical quadrature what gives me $0.424413181578394$ that has an error of $-6.827871601444713\times 10^{-15}$.

Nicoguaro (talk) 19:58, 22 November 2011 (UTC)

coherent vs. in-coherent

the article does not state weather the diffraction pattern is for coherent or incoherent illumination. also, it is not stated weather the maximum intensity calculation is for coherent or incoherent illumination.

the Airy disc pattern for the coherent case is just const*J(x)/x. without squaring the expression. i cant find the right intensity statement —Preceding unsigned comment added by 62.0.44.101 (talk) 07:07, 11 May 2010 (UTC)

Mathematical details mistake

There seems to be a mistake in the mathematical details section. Taking the derivative of P{\theta), \frac{dP}{dz}=2 \frac{J_1(z)^2}{z}. So somebody should change that. —Preceding unsigned comment added by 147.52.186.60 (talk) 15:13, 5 October 2010 (UTC)

Image useful?

The bottom image http://en.wikipedia.org/wiki/File:Spherical-aberration-slice.jpg is a very nice image, but since it mostly seems to be about showing the effects of spherical aberration, which is not discussed in the article, it could be questioned why it is included. I would suggest incorporating the middle image of this figure (the non-aberrated image, which does show the Airy pattern) in the article, and deleting the top and bottom. Geoffrey.landis (talk) 15:31, 21 January 2011 (UTC)

Definition of 'x' in section 'Approximation ...'

The symbol 'x' in the section on the Gaussian profile approximation can not be the same as the x introduced in the section 'mathematical details'. The x in the gaussian approximation section must have the physical dimension of length, while the x in the section on the mathematical details is dimensionless. I would guess from the figure caption in the gaussian approximation section that 'x' should be replaced by the off-axis coordinate 'q' defined earlier, but not entirely sure... --JocK (talk) 02:59, 10 October 2011 (UTC)

They're just two different uses of x as a formal variable, unrelated. Dicklyon (talk) 03:16, 10 October 2011 (UTC)

Sure. The issue is that this 'formal variable' is undefined. Have changed the text to give it the above definition. Change it if you don't agree. JocK (talk) 00:15, 30 October 2011 (UTC)

Yes, I think you did good. Dicklyon (talk) 20:54, 30 October 2011 (UTC)

Size as it pertains to digital camera resolution

This article is the first place I stopped when researching the topic of theoretical resolution of digital cameras, but I see some problems. First the derivation of the figure of 4 µm is very unclear. Additionally, such a derivation should be based on optimal, not typical, conditions; as a good photographer will use equipment to exploit its strengths and is more concerned with the performance limits than "typical use." A f1.4 aperture is much larger than one of f8 and so would seem to make for significantly better resolving power. Third, the math does not seem to take into account that in the case of digital photography, blue light is not sampled at the same spatial resolution as green light, which is much more important to the eye; and furthermore, the distance between sensor elements of like color is not the same as the element pitch. A 4000-element-wide sensor has only 2000 elements of a given color across its width, making the Airy resolving power only part of the equation. This is in contrast to film, which samples all wavelengths of light at roughly equal spatial resolutions. If digital (or any) photography is to be discussed in this article, I recommend it be done by someone who is versed in both physics and photography. — Preceding unsigned comment added by 184.153.114.71 (talk) 23:21, 7 January 2012 (UTC)

Indeed, it's quite complicated, as the Airy disk is only one of many mechanisms that blur and limit resolution. It gets combined with aberrations, anti-aliasing filter, and the area of pixel microlens, and then sampled in unequal pattrens, making it very hard to put numbers on things. And at f/1.4 it's almost certainly irrelevant, as it's hard to make a diffraction-limited f/1.4 lens; aberrations will dominate there. You have a good point about the comment in the article. If the Airy disk is 4 microns, you can probably win a bit by making pixels in a Bayer sensor as small as 2 microns; and at f/4, maybe even smaller. Better look for sources if you want to say anything useful, but it's hard to find much sensible about it in sources, so good luck. Dicklyon (talk) 01:53, 8 January 2012 (UTC)

It occurs to me that you're not going to find much freely available information on the subject. Digital camera technology is a jealously guarded arms race and its protection from the eyes of competitors is a billion dollar poker game. The Eastman Kodak bankruptcy problems are a good example. Trilobitealive (talk) 16:53, 8 January 2012 (UTC)

I just now fixed a long-standing glitch in this section. To resolve objects spaced at the Rayleigh criterion, you need two pixels per object: one for the object and one for the space between. The earlier wording implied one pixel per object, none for the spaces. I made a minor wording change to correct the error. The wording now implies 4 pixels per Airy disk diameter. The minimum requirement is actually 4.88 pixels per Airy disk diameter, to meet the Nyquist minimum of 2 pixels per cycle at the diffraction-limited cutoff frequency. 4.88 pixels per Airy disk diameter is the value that ends up getting used by Nikon in their explanation at https://www.microscopyu.com/tutorials/matching-camera-to-microscope-resolution . But it requires some math to ferret out that fact, so rather than complicate the discussion here in Airy disk I just noted that pixels smaller than 2 per radius (4 per diameter) would not give "significant" improvement. RikLittlefield (talk) 00:01, 25 August 2018 (UTC)

Conditions for Observation

That section contains the following text "and the radius a of the aperture is not too much larger than the wavelength λ of the light." That is a very poor description of the far-field condition as all optical systems have apertures many times larger than the wavelength of light. That part needs some educated editing.

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Deconvolution of Airy pattern

I believe it is possible to use deconvolution to improve the spatial (and thus angular) resolution of the image beyond that of the Airy disk. The task is not easy by itself, but it is technique used in microscopy and astronomy. The detector used must supersampling spatially, the model of the Airy pattern (or in general a point spread function, which also includes other effects) need to be computed theoretically or obtained from a real instrument, and complex computer methods (not simple inverse convolution method) need to be used to make it actually work under presence of multiple sources, noise and quantization and registration errors). One of the more feature full packages for astronomy is AIRY - https://www-n.oca.eu/caos/airy-article-SPIE.pdf , but even it doesn't actually contain all the methods used in research. ImageJ software also does have deconvolution methods and it is used often in microscopy and astronomy. There is number of other custom program to do similar jobs, often application specific. I think it would be useful to add a section about it, and put a some suitable link in the See also. 81.6.34.246 (talk) 23:54, 5 January 2020 (UTC)

Reduce Image

Too many in lead(intro) Greatder (talk) 11:44, 13 July 2022 (UTC)

"Orders of diffraction"

Can someone explain what this means in the context of an Airy disk? Referring to the photo of the red diffraction pattern. I can see that the rings are visible much further out than for the typical Airy beam. 98.156.185.48 (talk) 02:52, 26 September 2023 (UTC)

comment

I would rewrite

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 from the aperture where the diffracted light is observed (the screen distance) is large compared to the aperture size, and the radius of the aperture is not too much larger than the wavelength of the light.

as

The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a normal-incidence plane wave (no phase or intensity variations across the aperture), the screen distance (the distance from the aperture where the diffracted light is observed) is large compared to the aperture size, and the radius of the aperture is not too much larger than the wavelength of the light. 151.29.137.229 (talk) 16:10, 11 October 2023 (UTC)