However, the angle β is so small that it is fair to write cos(β) ≈ 1, such that tan(β) ≈ sin(β). In fact, it is so small that sin(α) ≈ sin(β), such that α ≈ β.

It should be noted that it is possible for a lens to resolve a smaller spot the pixels in an imaging medium. This is the case when airy disks focused by a lens cover only a fraction of a photosite. In this case, even if two highly resolved point light sources generate airy disks that merge over a single photosite, the end result will be the same... the sensor will only detect a single point light regardless of the aperture. The "diffraction limit" of such a sensor would be higher (say f/16) than for a sensor that is able to distinctly resolve both point light sources (which might be diffraction limited at f/8). It is also possible, and likely that point light sources will NOT be perfectly focused onto the center of a photosite. It is entirely plausible for an airy disk to be focused at the border between two photosites, or the junction of four photosites. In a black and white sensor or foveon sensor (stacked color sensels), that would only cause softening. In a color bayer sensor, where a square junction of 4 photosites will be capturing an alternating pattern of GRGB colors, as airy disk can affect the final color rendered by those four photosites as well as cause softening or improper resolution.

The reason why the fringes grow increasingly dimmer is that, as the order increases, so does the magnitude of the denominator. This trend ensures that the intensity declines as we move further from the central maximum. This is a graphical representation of the symmetric, dampening intensity of a typical diffraction pattern.

Rayleigh criterion

The value of α for the central maximum is 0. We calculate its intensity in the following way. As α approaches 0, so does sin {πdsin(α) /λ}. When we apply limits to the entire expression, we find that the intensity I is equal to I0. This is the maximum intensity and it is achieved when α = 0, or at the central maximum.

However, before we can understand how the slit diffracts the light, let’s make one thing clear. While light does experience interference, it is only conspicuous when the two sources of light are both monochromatic — emitting light of a single wavelength — and coherent — emitting identical waves of a constant phase difference. When the sources are incoherent or multi-chromatic — or worse, both (which is the case with white light) — the bands produced are indistinguishably muddled and not as uniform and distinct. These conditions also must be fulfilled to conspicuously demonstrate diffraction.

Now, there exists a value of α at which when two waves bend, they are rendered out of phase. These two waves negate each other or interfere destructively to produce a minimum – a region of darkness. Here, the peak of one wave is superimposed on the valley of the other.

Now, according to Huygens’ principle, every point between the edges is a source of waves. While the interference pattern studied above is formed due to the interference of two different waves emanating from two different slits, a diffraction pattern is formed due to the interference of different waves emanating from a single source. How is this possible?

There also exists a value of α at which, when two waves bend, they are rendered in phase with each other. These two waves add or interfere constructively to produce a maximum – a region of brightness. Here, the peak of one wave is superimposed on the peak of another. It is now obvious why the pattern is merely a bold bright spot when the slit is wider than the light’s wavelength. When the light simply falls through the wide slit, not a single wave bends. They all pass undeflected and therefore exist in the same phase. They all interfere constructively on the screen ahead.

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However, does the stream of light always flow around the obstacle? No, particularly when the obstacle is too large. A detailed understanding of the phenomenon of diffraction will reveal why this is the case.

Diffraction limit is the maximum sharpness limit of a lens due to laws of physics. Basically you cannot get any sharper photo no matter how many pixels your camera have or how perfect the optical system is.

The diffraction limit is the point where airy disks grow large enough that they begin to affect more than a single photosite. Another way to look at it is when the airy disks from two point light sources resolvable by the sensor begin to merge. At a wide aperture, two point light sources imaged by a sensor may only affect single neighboring photosites. When the aperture is stopped down, the airy disk generated by each point light source grows, to the point where the outer rings of each airy disk begin to merge. This is the point where a sensor is "diffraction limited", since individual point light sources no longer resolve to a single photosite...they are merging and covering more than one photosite. The point at which the center of each airy disk merges is the limit of resolution, and you will no longer be able to resolve any finer detail regardless of the aperture used. This is the diffraction cutoff frequency.

Now that we have discovered what determines the distance by which the fringes are separated, we can turn to the second question: Why is the central maximum the brightest, while the subordinate maximums become increasingly dimmer?

The value of sin(α) for minimums is ±nλ/d. When we substitute this in the expression, we find that the numerator is reduced to sin(nπ), which is equal to zero, precisely what we were expecting. Now, the value of sin(α) for maximums is equal to ±(n+1/2)λ/d. This is because the waves that interfere constructively travel the same distance that the waves interfering destructively do. However, they — as one can infer from the diagram — also travel an additional 0.5 λ/d. Basically, they are only approximately halfway between the minimums.

Diffractionlimit calculator

Whoa, Stop right there: Larger numerical apertures produce smaller spots makes sense if you consider that in the formula, the aperture is used as a reciprocal value. Dispersion also plays a role here, too.

Whubers sequence of images above is a decent example of the effect of diffraction, as well as the effect of optical aberrations when the lens is wide open. I think it suffers a bit from some focus shift due to spherical aberration, so I have created an animated GIF that demonstrates the effects of changing the aperture of a Canon 50mm f/1.4 lens down from its widest aperture to its narrowest, in full stops.

There have been some very good answers, however there are a couple details that have not been mentioned. First, diffraction always happens, at every aperture, as light bends around the edges of the diaphragm and creates an "Airy Disk". The size of the airy disk, and the proportion of the disk that comprises the outer rings, and the amplitude of each wave in the outer rings, increases as the aperture is stopped down (the physical aperture gets smaller.) When you approach photography in the way Whuber mentioned in his answer:

Gaussian beam

The undesired effect is when you are setting zoom larger than the diffraction limit allows, and a photo is not getting sharper, only larger. This often happens in telescopes and microscopes. This is also why electronic microscopes are used instead of optical, as the optical ones cannot see clearly sharper than X.

Now, waves that are not equidistant from the center — say, the waves generated by the first point and the point just below the center of the slit – on their journey to the screen, don’t travel an equal distance. One can see from the diagram that a pair of waves is out of phase when one lags the other by half the light’s wavelength. The waves then interfere destructively to produce a minimum. This would also be true for the second point and the point just below the center. One can discern a pattern – the angles at which minimums are generated.

The rules are the same: the light bends around the slit, which causes the waves to deflect as illustrated above. This renders a few in phase and others out of phase with each other. The waves then interfere constructively and destructively to produce – because they are incoherent and multi-chromatic — a variegated pattern of colors. Refer again to the expression we derived for sin(α). A diffraction grating obeys the same laws. We know that sin(α) is proportional to the wavelength λ of the diffracted light. Therefore, for slits of equal width d, red light is deflected more severely than blue light, as the former’s wavelength is much longer.

My Canon 450D, a 12.2mp APS-C sensor, has a diffraction limit of f/8.4. In contrast, the Canon 5D Mark II, a 21.1mp Full Frame sensor, has a diffraction limit of f/10.3. The larger sensor, despite having nearly twice as many megapixels, can go an extra stop before it encounters its diffraction limit. This is because the physical size of the photosites on the 5D II are larger than those on the 450D. (A good example of one of the numerous benefits of larger sensors.)

Diffraction is the bending of waves around an obstacle. A diffraction grating is an obstacle with many slits that diffracts waves in a particular pattern.

Waves that bend at an angle that satisfies this equation interfere destructively. Here, n is the integer that represents the order of the minimum. The first minimums are produced on both sides when sinα = ±λ/d. The second-order minimums are produced on both sides when sinα = ±2λ/d, and so on. Between every minimum is a maximum. Lastly, at n = 0, the central maximum is produced where one would expect a minimum. This hypothetical minimum is flanked by two maximums – this is why the width of the central maximum is twice that of the other maximums. In terms of sin(α), it is 2λ/d.

You can measure this reasonably well with your own lenses and no special equipment. Mount the camera on a tripod in front of a sharp, detailed, well-lit flat target having lots of contrast. (I used a page from a magazine; it worked fine.) Use your best settings: lowest ISO, proper exposure, mirror locked up, medium focal length for a zoom lens (or vary the focal length, too), middle distance, perfectly in focus, RAW format. Take a series of photos in which you vary only the f/stop and the exposure time (to keep the exposure constant). Look at the sequence of pictures at 100% on a good monitor: you will see where your camera's "sweet spot" is and you will see the effects of using wider or narrower apertures.

The point at which the lens is stopped down as far as it can go without increasing softness is the diffraction limit. On some lenses, that's as far as you can stop down -- Nikon, for example, has traditionally kept a relatively wide minimum aperture (f/16) on many of their designs. On other lenses (macros, especially) you might still have a couple of stops or more available to you; depth of field considerations may be more important than absolute sharpness in some applications.

What’s wonderful is that the pattern can be generated with a single slit as well. However, in a pattern generated by a single slit, unlike the pattern generated by two slits, the intensity of light is not evenly distributed. The pattern is called a diffraction pattern, because the light with which it is painted is diffracted.

You may often come across tables on the internet that specify a specific diffraction limited aperture for specific formats. I often see f/16 used for APS-C sensors, and f/22 for Full Frame. In the digital world, these numbers are generally useless. The diffraction limiting aperture (DLA) is ultimately a function of the relation of the size of a focused point of light (including the airy disk pattern) to the size of a single light sensing element on a sensor. For any given sensor size, APS-C or Full Frame, the diffraction limit will change depending on the size of the photosites. An example of this can be seen with Canon's EOS Rebel line of cameras over the years:

Think of a scene as comprised of many small discrete points of light. A lens is supposed to convert each point to another point at an appropriate place on the image. Diffraction causes every point to spread in a circular wave-like pattern, the Airy disk. The diameter of the disk is directly proportional to the f-number: that's the "diffraction limit."

Diffraction, along with interference and polarization, is an indisputable proof of the wave nature of light. It is diffraction that makes the light radiated by a source detectable, even when its path is obstructed by an obstacle. The light, like water, flows around the obstacle to reach our eyes. Diffraction is why we can detect a source that is situated beyond the curve or why the edges of a cloud obscuring the Sun still gleam, accentuating what we call its silver lining.

I've seen the term used, but what is a "diffraction limit", when should I worry about it, and what undesirable effects are a result of it ?

The story should be similar for film grain size. Films with finer grain would ultimately be more susceptible to diffraction softening at lower apertures than films with larger grains.

It should also be clearly noted that the diffraction limit is not actually a limitation of a lens. As noted above, lenses are always creating a diffraction pattern, only the degree and extent of that pattern changes as the lens is stopped down. The "limit" of diffraction is a function of the imaging medium. A sensor with smaller photosites, or film with smaller grain, will have a lower limit of diffraction than those with larger photosites/grains. This is due to the fact that a smaller photosite covers less of the airy disk area than a larger photosite. When the airy disk grows in size and intensity as a lens is stopped down, the airy disk affects neighboring photosites.

Diffraction happens. It's a fact of life. When lenses are used wide open, other lens abberations are far too prominent for you to notice a minor sharpness loss due to diffraction. Stop down a little bit, and those abberations are minimised -- the lens seems to just get better and better. Diffraction is there, but you still don't really notice it because light that is not passing near the edges significantly outvotes the light that is passing getting a little too close to the aperture blades.

Airy disk formula

The secondary waves that emanate from these point sources interfere with each other as they bend around the slit. This is because the bending causes a wave to travel a longer distance than another wave. Let’s assume that the parallel rays bend at the slit at an angle α.

The slit and the screen are separated by a distance D, the magnitude of which is enormous compared to the slit’s meager width d. The angle drawn between two waves producing the first minimum on the screen is β. The first minimum is situated at a distance y(1) from the axis. Observe that:

At some point while you are stopping the lens down, the gains you make by eliminating the optical differences between the center and outer parts of the lens elements starts to go away -- there is no longer enough crisply focused light to drown out the out-of-focus image caused by light bending around the edges of the optical path (diffraction). The lens isn't going to get any better when you stop down anymore -- too much light is being diffracted compared to the light that's getting through the middle. From this point on, stopping down will make the image softer.

Diffractionlimit

When you have reached the diffraction limit of your camera, ANY lens stopped beyond that aperture will give you softer results. It is directly related to the size of individual pixels, not the sensor size.

Let’s redraw the diagram, illustrating the deflection of waves above. To reach the center of the axis ahead, waves generated by points equidistant from the center of the slit — say, the first and the last point — must travel an equal distance, that is, such a pair of waves is in phase. This is why the central maximum is bright – it is formed by waves that have traveled an equal distance and are therefore in phase and have interfered constructively.

Abbediffractionlimit

Diffraction is often touted as an image killer, and people talk about the "diffraction limit" as the point at which you can no longer resolve an image "usefully". On the contrary, the diffraction limit is only the point where diffraction starts to affect an image for the particular image medium you are using. The diffraction cutoff frequency is the point at which additional sharpness is impossible for a given aperture, and this is indeed a function of the lens and physical aperture.

This states that the reciprocal of the wavelength \$\lambda\$ of the light being focused multiplied by the f-number \$N\$ of the lens is the number of cycles per millimeter that can be resolved. The diffraction cutoff frequency is generally the point where resolution reaches the wavelength of the frequencies of light itself. For visible light, λ between 380–750nm, or 0.38–0.75 microns. Until the cutoff frequency has been met for a given aperture, more resolution can be achieved.

(Note: The image is large, 3.8meg, so let it fully download to see the comparison of sharpness at each stop.) The image exhibits marked optical aberration when shot wide open, particularly Chromatic Aberration and some Spherical Aberration (there may be some slight purple fringing... I tried to get focus dead on.) Stopped down to f/2, CA is lessened considerably. From f/2.8 through f/8, sharpness is at its prime, with f/8 being ideal. At f/11, sharpness drops ever so slightly, due to diffraction. At f/16 and particularly f/22, diffraction visibly affects image sharpness. Note that even with diffraction blurring, f/22 is still considerably sharper than f/1.4 or f/2.

On modern DSLRs, the diffraction limit will be hit between F/11 and F/16. On cameras with small sensors, it may be F/8 or even less. You'll notice that most tiny cameras do not use apertures smaller than F/8 for this very reason. Some even use a fixed aperture (F/3.5 or so) and simulate less light coming in by slipping a ND filter instead of stopping-down. Unfortunately, they actually put the simulated F-stop in the EXIF, so you need to know the camera to realize it uses an ND filter rather than a normal aperture.

The following sequence is taken from a series for the Canon 85 mm f/1.8 lens, which is a pretty good one. From top to bottom are 100% crops (converted to high-quality JPEG for Web display) at f/1.8, 2.8, 5.6, 11, and 22. You can see the increasing effects of diffraction at f/11 and f/22 in the bottom two images. Note that for this particular lens used with this particular camera (EOS T2i, an APS-C sensor), the diffraction softness at high f-numbers doesn't approach the softness seen with the lens wide open. Having comparable information for your own lenses, which can be obtained in a few minutes, can be valuable for choosing exposure parameters in important photos.

The wavelengths, from blue to red, bend increasingly. The grating therefore splits the white light just how a prism does, and what is dispersed is a splendid, iridescent rainbow on the surface of the CD.

Lastly, such a symmetric pattern is produced when the light is monochromatic and coherent. When white light — a medley of wavelengths exhibiting tremendous incoherence — is diffracted, the pattern generated is profoundly variegated. This is evident on CDs as a vague, hazy rainbow.

To calculate the intensity of the first or the rest of the maximums, substitute in the expression, sin(α) = (n+1/2)λ/d, where the value of n is the order of the maximum whose intensity you wish to calculate. We find that the intensity of the, say, first-order maximum, is equal to 4I0/9π2, or 0.045I0. This is a huge dip in magnitude, but it is imperceptible to the human eye.

All of photography is a compromise. There may be times when you want to stop down farther than the optimum, but it helps to be aware of the compromises you're making. Stopping down is an easy answer to DOF, but if you're hooked on landscapes and taking them all at f/22 or f/32, it may be time to take a look at a tilt/shift lens.

A century later, it was the British polymath Thomas Young who successfully demonstrated how light behaves like ripples in a pond by forcing light to squeeze through two adjacent slits. What the squeezed lights illuminated on a screen ahead of them is now called an interference pattern – a uniform, alternating pattern of bright and dark bands.

Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

Numerical aperture

A CD is composed of extremely thin, equally distant parallel wires. When it is illuminated, the gaps act as slits. The width of every slit is comparable, even smaller than the wavelength of light, and therefore, every slit naturally diffracts the light. In the field of optics, such a series of extremely thin, equally distant parallel wires is called a diffraction grating.

While the two conditions must be fulfilled to ensure that the phenomenon is observable, there exists another condition which, if not fulfilled, prevents the phenomenon from occurring in the first place. A diffraction pattern is produced only if the wavelength of light λ is comparable to or larger than the size of the obstacle around which it will flow. If the width d of the slit is very large — similar to how a needle would fall in a slot for coins — the light would simply pass untouched, and a single bright spot would be illuminated on the screen ahead. However, when the slit is narrow, the light is diffracted spectacularly.

While the answers already here describe diffraction well. Diffraction limit is most often used to describe the point at which stopping down your lens does not give you more details in relation to the pixel-size of your camera's sensor.

F-number

The spot in the center of the axis, spanning equal lengths on both sides, is the brightest. This is the central maximum. It is flanked on both sides by the first-order minimums, which are followed by the first-order maximums, which are followed by the second-order minimums and so on, alternatively. Even though the disparity is imperceptible to the human eye, the intensity of maximums decreases as we move farther from the central maximum. What determines the distance by which the maximums or minimums are separated? And what determines the pattern’s intensity? Let’s find out.

The implication is that if the screen distance D and the wavelength y are constant, the distance y increases or the pattern gets wider as the slit gets narrower. This is why, when the slit is narrow, the light is diffracted so spectacularly.

As the f-number is increased from its minimum (a wide open lens), the light falling at a point on the image will come from a narrower region of the lens. That tends to make the image sharper. As the f-number is increased, the Airy disks get larger. At some point the two effects balance to make the sharpest image. This point is typically in the f/5.6 to f/8 range on SLR cameras. With smaller f-numbers, overall properties of the lens (its aberrations) take over to make a softer image. With larger f-numbers, the softness is dominated by the diffraction effect.

Contrary to Newton’s belief, Christiaan Huygens, in the 17th century, suggested that light doesn’t behave like a particle, but rather like a wave. He postulated what is now called Huygens’ principle: every point on a wave of light is a source of secondary waves that travel at the same speed as light. He also elegantly explained the occurrence of optical phenomena, such as reflection and refraction, with his wave theory of light. However, Huygens could never demonstrate the wave nature of light. He failed to prove his claims experimentally.

This page at the Cambridge In Color site has a detailed technical explanation of the diffraction limit. It's also got an on-line calculator for checking whether a particular combination of aperture, camera, print size, and viewing distance is diffraction-limited or not.

You realize that every one of those points of light, when focused by your lens, is generating its own airy disk on the imaging medium.

The discovery vindicated Huygens, as light cannot bend or flow around an obstacle unless it obeys his principle. Only waves interfering with each other can form such a pattern. Young immediately realized that when the two waves are squeezed between the slits, a bright band is produced when the peak of one ripple interferes constructively with or is added to the peak of another ripple, while a dark band is produced when the peak of a ripple interferes destructively with or negates another ripple. The addition causes the luminosity of the region to double, whereas the negation renders the region utterly dark.

Arm-waving: Lenses can focus light to a small spot but not a point. The spot size can vary with the wavelength, with short wavelengths forming smaller spot sizes than longer ones. When a very good, aberration-free (diffraction-limited) lens is used, collimated light will produce an airy disk as a spot at the focus. An airy disk is still the smallest spot that can be produced with that lens at that aperture with that wavelength (using collimated light). Larger apertures produce smaller spot sizes with tighter focus and reduced depth of focus than smaller apertures.

As already explained, when the waves of light encounter the slit, they bend and squeeze through, like running water suddenly spouting out from a crack in the pipe. As the waves bend and change direction, they appear to spread and mimic ripples. The ripples can be approximated with parallel lines. Why? Because the screen is so far away that the waves appear to be straight lines, just how currents in the Nile would be imperceptible from the International Space Station.