Diffractionlimit formula

Materials that produce light instantly are called fluorescent. The atoms inside them absorb energy and become “excited”. While returning to the normal state in approx. a hundred thousandth of a second (10-9 to 10-6 sec), they release the energy as tiny particles of light called photons.

All camera optics are plagued by twin demons of interference and diffraction. These yield stray light rays that comingle with the image forming rays. Diffraction is caused when light rays from the vista being imaged, just brush by the edge of the aperture stop. Some close passes are shadowed but not completely blocked. The ricochets comingle and degrade the image. Interference is due to the wave nature of light crossing paths and adding and canceling each other.

The size and nature of the Airy disk is not something you can overcome — it's a function of the wave-like behavior of light, the aperture size (usually assumed to be circular), and the wavelength of the particular light in question).

Diffraction limited systemapplications

What we often refer to as the Diffraction Limited Aperture (DLA) for a specific digital sensor is the aperture at which the effects of diffraction are noticeable when the resulting image file is viewed at a magnification that yields 1-pixel in the image file equals 1-pixel on the monitor and those individual pixels are right at the limits of the viewer's perception to differentiate them. The DLA is the point at which the effects of diffraction are barely perceptible at such a magnification. This begins to occur when the size of the blur caused by diffraction becomes larger than the size of a digital camera's sensor pixels.

Technically speaking, fluorescence is a radiative mechanism by which excited electrons transition from the lowest excited state (S1) to the ground state (S0). During this process the electron losses a bit of its energy by vibrational relaxation, resulting in the emitted photon having lower energy and therefore longer wavelength.

Yes but you are trading one type of information for another - it doesn't break the laws of physics or information theory. You have to assume the object is stationary and you are trading signal to noise for resolution.

Sort of, to a limited degree. Using sub-pixel shifting of the imaging sensor, in effect you are increasing each pixel size while keeping their spacing the same. Of course, it is not physically possible to build sensors where individual pixels are larger than their pitch (center-to-center spacing). But mathematically, this is basically what's happening.

The resolution limits are extensively studied in microscopy, so it is useful to look at the techniques developed there. However, all the techniques that provide resolution beyond the diffraction limit are not applicable to form a natural color image and without altering the subject.

Well, as you mentioned, it requires a non-moving subject, that's one of the limits of applicability. As John stated in his answer, you are using the temporal-based certainty (i.e., there is no motion in the scene, so it exists independent of time) to take multiple images (which takes time, but who cares, you have plenty of it when the subject isn't moving) that help you increase your spatial information / knowledge about the scene.

Common super-resolution techniques that can be used in this setting today try to increase the resolution by recovering the original image that we can only sub-sample with our limited sensor resolution (combining multiple slightly offset images) and/or modelling the imperfections of the optical system that prohibit it from reaching the diffraction limit. Another concept worth mentioning here is apodization, but this does not overcome the diffraction limit but merely removes the non-central maxima of the Airy disk.

Basically, there are 3 main forms of luminescence: fluorescence, phosphorescence and chemiluminescence. Two of these, namely fluorescence and phosphorescence, are forms of photoluminescence. The difference between photo- and chemiluminescence is that in photoluminescence the luminescence reaction is triggered by light whereas in chemiluminescence the light emission is triggered by a chemical reaction. The basis of both forms, fluorescence and phosphorescence, is the ability of a substance to absorb light and then emit this light at a longer wavelength which means lower energy, just the timescale in which that happens is different. While in fluorescent reactions the emission takes place immediately and is only visible as long as the light source is on (e.g. UV lights), in phosphorescent reactions the material can store the absorbed energy and release it later, resulting in an afterglow that persists after the light has been switched off. So, if it disappears immediately, it’s fluorescence. If it lingers, it’s phosphorescence. And if it needs chemical activation, it’s chemiluminescence.

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Diffraction limited systemwikipedia

Enter pixel-shift. With pixel shift, even though the individual pixels stay the same and each individual pixels spectral transform remains the same, taking multiple shots of the same subject with sub-pixel shifts allows us to decrease the pixel pitch beyond the physical distances. If we halve the pixel pitch, we double the aliasing frequency.

UPDATE Thank you for your answers. However it feels to me more like you explained to me what diffraction means rather than if it is possible to overcome the diffraction limit under reasonable assumptions. To clarify further: In a relatively controlled environment, where you can expect the subject to be stationary and the lens/aperture diffraction to be the limiting factor of resolution (as opposed to sensor resolution), do techniques exist to increase detail beyond this diffraction limit without the aforementioned "special assumptions"?

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Considering the 2D case, we also have to look at the shape of the aperture and the pixels - to say it in the words of Sepp Herberger: "The round must go in the square", where the round is the image of the aperture - the airy disk and the square is the pixels. But for the moment, this will be left as an excercise for the reader ;)

Several post-processing techniques that increase the resolution limits of a camera/lens system can be used to ameliorate the effects of diffraction. Stacking multiple images taken from slightly different positions as you suggest is one way. A tool such as Canon's Digital Lens Optimizer that uses very detailed lens profiles is another.

Diffraction limited systemformula

We all think of something glowing when we hear the words luminescence, fluorescence or phosphorescence but the mechanism that stand behind these effects are somehow different. So, the question is, what sets them apart from each other?

For more on how the DLA is affected by the resolution limits of the recording medium, please see: Does sensor size impact the diffraction limit of a lens?

Looking at phosphorescence we need to take a short detour into electron spin to understand the differences between fluorescence and phosphorescence. Spin is a fundamental property of an electron and a form of angular momentum that defines behavior in an electromagnetic field. The spin can only have a value of ½ and an orientation of either up or down. An electron’s spin is therefore designated as +½ or -½, or alternatively as ↑ or↓. If electrons are on the same orbital of an atom, they always have an antiparallel spin at single ground state (S0). When promoted into an exited state, the electron maintains its spin orientation and a singlet excited state (S1) is formed, where the both spin orientations remain paired as antiparallel. All relaxation events in fluorescence are spin neutral and the spin orientation of the electron is maintained at all times.

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As an example, you could imagine a night club where the fabric and teeth glow under the black light (fluorescence), the emergency exit sign glows (phosphorescence) and also the glow sticks glow (chemiluminescence).

But we can get more information on the shape if we create multiple exposures that are ever-so-slightly shifted, effectively creating virtual pixels of a smaller size when we apply our math appropriately.

As Michael Clark stated in his answer, a camera system is diffraction limited when the size of the Airy disk (the blur) caused by diffraction becomes larger than the size of a digital camera's sensor pixels.

Diffraction limited systempdf

In phosphorescence this is different. Here you have fast (10-11 to 10-6 sec) intersystem crossings from singlet excited state (S1) to an energetically favorable triplet excited state (T1). This leads to inversion of the electron spin and these states are characterized by parallel spin of both electrons and are metastable. Here the relaxation occurs by phosphorescence, resulting in another flip of the electron spin and the emission of a photon. The return to the relaxed singlet ground state (S0) might occur after a longer delay (10-3 to >100 sec). In this process more energy is consumed by non-radiative processes during phosphorescent relaxation than in fluorescence, leading to a higher energy difference between the absorbed and emitted photon and therefore a bigger shift in wavelength.

For an Airy disk that results from a given aperture we can take a look at two extreme cases. In the first case we have a sensor that has a significantly higher pixel pitch than the size of the Airy disk. In this case, we severely subsample the Airy disk and only know that there is a blob of light hitting a single pixel, but nothing about its shape.

As far as I know for color images, the only possible improvement to the size of the Airy disk for a given lens aperture is in the special situation of transparencies - the size of the airy disk can be decreased by a factor of 2 with a special lighting setup and a bit further using oil immersion - these techniques alter the numerical aperture of the whole optical system, instead of just the lenses'.

His calculations, remain valid. We are talking about the resolving power of a lens system. Following is a table for 589, about the center of our color spectrum.

Even with an optically perfect lens, the resolution will always be limited by the Rayleigh Criterion as pointed out by the other answers (if no other limits like e.g. lens imperfections lower our resolution further).

There are many possible approaches. One is simply blocking out the centre of your optical system and only using the edges. The central peak of the transfer function of this is narrower than for a circular aperture so your resolution is increased but you both have less signal received and have wider wings in the transfer function, both reducing signal to noise.

Diffraction limitedtelescope

"Without making special assumptions" in this case means the techniques of superresolution microscopy - structured light, laser beams etc.

What we refer to as the Diffraction Cutoff Frequency requires a much narrower aperture setting than the DLA for a specific sensor (or film - the size of the grains in various films affects the DLA with film!).

Now add the sensor to the equation. The sensor consists of many rectangular photosites and for modern image sensors let's assume pixel pitch equals pixel size (e. g. 100% fill factor). Each of these squares has a frequency spectrum of $$|F(f)| = \frac{1}{X_{pixel}}sinc(f \frac{1}{X_{pixel}})|$$. However, the spatial frequencies that lie beyond half the pixel pitch will become visible as aliasing in the image - to avoid this, we need to shrink the distance of the pixels down to the point that sufficiently much of the information in the airy spectrum remains in the pass-band of the sensor. Practically however we cannot create pixel pitches smaller than the pixels themselves. They cannot overlap, right?

To illustrate the point: Can superresolution beyond the diffraction limit be achieved by taking multiple exposures from slightly different angles and positions and feeding them into [SR approach here]? Even with the added assumption of a diffraction-limited system (High resolution camera and lens)?

diffraction-limitedspot size formula

So in essence, the frequencies present behind the lens are the result of the interplay between the airy disk given as a function of f-number, the photosite size and the pixel pitch $$ sinc^2(\mathrm{aperture}), sinc(X_{\mathrm{pixel}}), DiracComb(X_{\mathrm{pitch}})$$.

However consider now that we massively increase our sensor resolution (or we can pixel shift to that effect), then we sample the Airy disk so much that we can perfectly reconstruct its true shape. We cannot know however what the true shape of the object is that creates the Airy disk - it might be as small as a single nanometer-sized photon source, but its image is never smaller than this blob of a few microns in diameter.

Abbediffractionlimit derivation

Passing this through an ideal but finite-sized aperture leads to a low pass filtering - the single point gets projected onto a less sharp image we call the Airy disk - whose frequency spectrum follows the square of the sinc function (|F(f)| ∝ sinc^2(f)). So this attenuation of higher frequencies is the fundamental limit in resolution for a given optical setup.

(Note: This section still needs work in terms of readability and clarity, but it also has become mostly redundant after finding the wikipedia article on optical resolution)

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Diffraction is a lot like the edges of depth of field. The more we magnify an image the easier it is to see. Diffraction starts at apertures where only very high magnification will reveal any effects at all. As the aperture is closed down further the effects begin to be perceptible at lower and lower magnifications.

(Sidenote: With today's fill factors of close to 100%, the achievable resolution increase is limited. This is because we are sampling with a rectangular window instead of a single point, but I couldn't find an authoritative source on the limit and have not worked out the math myself yet.) See below - Theoretical foundations

Every point in the photographed scene can be understood as a single dirac pulse - from a spectral view, a dirac pulse contains all frequencies equally (|F(f)| =const.).

But if you can increase the size of the pixels while still packing the same number of pixels in the same area, you can "push back" the diffraction limit a bit farther. And that's what sub-pixel shifting of the image sensor does.

As the title says, can the diffraction limit be overcome with superresolution techniques? Or is this the absolute hard limit in optical photography without making special assumptions? If this is the hard limit, what is an illustration of why this is the case?

The word luminous basically means giving off light. Most objects in our world give off light because they are in possession of energy that originated from the sun, the most luminous object we know and that we can see. In contrast to the moon who seems to give off light, but is only reflecting it from the sun like a giant mirror made of rock.