Point a perfect lens at a single distant star and it still cannot draw a perfect point. The best it can do is a small bright disk with faint rings around it. That pattern is the Airy disk, and it appears even when the lens is flawless, focused, and perfectly aligned.

The Airy disk is the fingerprint of diffraction through a circular aperture. It has a definite size, and that size puts a floor under sharpness: two details can sit close enough that their disks overlap before the image separates them. This page builds that idea from the ground up, with a simulator you can drag, the function that describes the pattern, and the way astronomers, microscopists, and photographers read it.

The preview places the aperture on the left and the resulting spot on the right. Drag the orange handle on the aperture rim and watch both change. Widening the opening pulls the spot into a tighter core; narrowing it spreads the pattern out. A larger aperture does gather more light, but that is only half the story. The opening also sets the shape of the smallest spot the lens can make.

Everything that follows makes that sentence precise. A finite aperture behaves like a row of tiny wave sources. Every point across the opening launches a wavelet, and the wavelets arrive at the focal plane with slightly different travel distances. Where they line up in phase they reinforce, and where they fall out of phase they cancel. The bright center is where the whole aperture reinforces at once, and the rings are where cancellation wins.

What the Airy Disk and Airy Pattern Are

Start with the simplest subject: one distant point of light. A ray diagram draws every ray from that point meeting at a single perfect image point. That model places the focus correctly, because a lens bends light by refraction and Snell’s law, but it leaves out diffraction. Diffraction is what happens when a wave is limited by an opening or an edge. The opening cuts down the wavefront, and the surviving parts of the wave recombine after the aperture.

For a circular opening, that recombination has a distinctive shape. Most of the light lands in a bright central spot. A smaller amount lands in faint rings around it. The whole spot-and-ring figure is the Airy pattern, and the bright central region is the Airy disk. This circular-aperture case is sometimes called Airy diffraction, after George Biddell Airy’s 1835 paper, “On the Diffraction of an Object-glass with Circular Aperture”.

The first correction to the usual “better lens means smaller dot” story belongs here. Even a perfect circular lens has a smallest possible spot. Better manufacturing can remove defocus, astigmatism, coma, and other defects, but it cannot remove the diffraction pattern created by the aperture. Optical engineers call the result the diffraction-limited spot: the limit that remains when the lens is otherwise ideal.

That spot is not an arbitrary blur. Its radius out to the first dark ring has a predictable form:

$$r_1 = 1.22 \frac{\lambda f}{D}$$

Each symbol maps to something physical:

• $\lambda$ is the wavelength of the light.
• $f$ is the focal length of the lens.
• $D$ is the diameter of the aperture.
• $r_1$ is the radius from the center out to the first dark ring.

Three practical rules fall out of it. Longer wavelength makes the spot larger. Longer focal length makes the spot larger. A wider aperture makes the spot smaller. Those rules are why blue light resolves slightly finer detail than red light in the same system, and why a wider telescope can split closer double stars.

Photographers usually describe the aperture through the f-number $\mathrm{N} = f/D$, so the same relationship also reads:

$$r_1 = 1.22 \lambda N$$

At a high f-number such as f/16, the aperture is small compared with the focal length and the diffraction spot is large. At a low f-number such as f/2.8, the spot is small. That is the tension behind a familiar camera decision: stopping down can reduce lens aberrations and deepen the depth of field, but it eventually enlarges the diffraction spot and softens fine detail.

From the microscope side, shorter wavelengths and larger numerical apertures shrink the Airy disk, while longer wavelengths and smaller apertures widen it. Cameras, telescopes, and microscopes use different units, but the tradeoff is identical.

Airy Disk Simulator: Wavelength, Aperture, and Focal Length

The simulator shows the sensor-plane spot on the top left and its radial profile on the top right. The profile is brightness versus distance from the center, and the orange line marks the first dark ring, the $r_1$ distance from the formula above.

Three draggable handles sit in the optical bench along the bottom. The aperture blade opens and closes the stop, the sensor plane changes the focal distance, and the spectrum marker changes the wavelength. Because all three feed the same diffraction scale, any one of them resizes both the spot and the profile.

Dragging the aperture open pulls the orange marker inward, so the core shrinks. Moving the marker to a longer wavelength or dragging the sensor farther out pushes it outward. Those are the three levers in the formula, and the simulator is a way to feel how they multiply rather than memorize which way each one moves.

The takeaway is that diffraction blur is structured, not random. Its size follows wavelength, aperture, and focal length, and the next question is how much those small patterns overlap.

The Airy Disk Function

The radial profile is a specific function, not a generic soft blob. For an ideal circular aperture, the intensity at radius $r$ from the center is:

$$I(r) = I_0 \left[\frac{2J_1(x)}{x}\right]^2, \qquad x = \frac{\pi D r}{\lambda f}$$

$J_1$ is the first-order Bessel function. It starts at zero, rises to a peak, and then oscillates while its amplitude decays. Dividing by $x$ and squaring converts that oscillation into the bright center and the alternating dark and bright rings. The first zero of the Bessel numerator falls at $x \approx 3.83$, which is exactly where the first dark ring sits. Rearranged, that zero gives $r_1 = 1.22\lambda f/D$, so the function and the radius formula are the same fact in two forms.

The shape also explains where the energy goes. Each ring is a radius where wavelets from different parts of the aperture cancel more than they reinforce, and the central disk is the region where they reinforce most strongly. About 84 percent of the light falls inside the first dark ring, and the faint outer rings share the remainder.

Because this function describes what happens to one ideal point of light, it is also called the point source function, or point spread function, of the circular aperture. The two names look at the same object from opposite ends: the Airy disk is what a single point looks like, and the point spread function is the rule that spreads every point in the scene.

Airy Disk Size: Ring Radius, FWHM, and Encircled Energy

Different fields quote different widths for the same Airy disk, so it helps to be explicit. FWHM is short for full width at half maximum. The three common measures are:

• The first dark ring, with radius $r_1 = 1.22\lambda N$ and diameter $2.44\lambda N$.
• The full width at half maximum, the width of the bright core at the height where intensity has dropped to half its peak. For an Airy disk this is about $1.03\lambda N$, noticeably narrower than the first dark ring diameter.
• The encircled-energy width, the diameter that contains a stated fraction of the light, such as the 84 percent inside the first dark ring.

When a specification quotes the full resolution of an Airy disk, it is usually naming one of these. Astronomy and microscopy tend to use the FWHM, optics textbooks tend to use the first dark ring, and instrument design tends to use encircled energy. Knowing which one is meant prevents a factor-of-two error when two systems are compared.

Airy Pattern and Optical Alignment

A single point source is an excellent alignment target, because a well-aligned optical system produces a clean, centered, circularly symmetric Airy pattern. That is the basis of the star test for collimating telescopes and of the visual checks used to center a laser beam or align a microscope condenser. Point the system at a point source, look at the focused spot, and judge its symmetry rather than its size.

When the optics are aligned, the core is round and the rings form even concentric circles around it. When something is off, the pattern loses symmetry in a way that hints at the cause:

• Coma, from a tilted or decentered element, pushes energy to one side and turns the core into a one-sided flare or teardrop.
• Astigmatism, where the lens focuses differently in two directions, stretches the spot into an ellipse on one side of focus and rotates that ellipse by 90 degrees on the other side.
• Defocus keeps the spot symmetric but spreads it into a larger set of concentric rings, so symmetry alone does not prove perfect focus.
• Aperture clipping or a decentered stop trims the rings on one side and can leave a bright edge.

Alignment procedures often watch the rings rather than the core. The core saturates easily and is too bright to judge, while ring asymmetry is easier to see. A centered first ring with an even halo is a practical sign that the aperture is being filled symmetrically and the axis is straight.

The important point is that the Airy pattern comes from the finite aperture and is present even in a perfectly aligned system. Alignment does not create the spot; it preserves the symmetry the diffraction pattern would have on its own. A misaligned system still forms an Airy-like pattern, but it spreads the light unevenly around the core.

Two Airy Disks Overlap: Rayleigh Resolution

One point source makes one Airy pattern, so two point sources make two Airy patterns. Far apart, the image shows two separate bright spots. Close together, the patterns add and can look like a single wider spot.

That is what resolution means here. Resolution is about whether the system preserves enough separation and contrast to distinguish two sources. Magnifying the image until it looks larger does not change that. The sources might be stars in astronomy, fluorescent emitters in microscopy, or tiny highlights on a manufactured part in machine vision.

A standard reference is Rayleigh’s criterion:

$$\theta_R = 1.22\frac{\lambda}{D}$$

This version describes angular separation: how far apart two distant points must appear, as seen from the lens, before the aperture can separate them. The angle shrinks when the wavelength is shorter or the aperture is larger, which is why a big telescope splits closer double stars than a small one at the same wavelength.

Rayleigh’s rule has a simple visual meaning. Two equally bright points count as just resolved when the center of one Airy pattern lands on the first dark ring of the other. At that spacing the combined profile has two peaks with a shallow dip between them, just enough to read as two sources.

The explorer plots the sum of the two patterns, with the individual patterns drawn faintly underneath. Drag either point marker in the star image toward the other and watch the profile. Below the Rayleigh angle the two peaks merge into one broad hump; near the limit the valley between them stays shallow; farther apart the peaks separate cleanly.

The aperture rim and the spectrum marker change the rule itself, while the point markers change the scene. A larger aperture lowers $\theta_R$, so the same pair becomes easier to resolve. A longer wavelength raises $\theta_R$, so a pair that was just resolved begins to merge. One changes the ruler, the other changes what is being measured, and it is worth keeping them distinct.

Rayleigh’s value is a stable reference point rather than a hard switch in nature. Detection also depends on contrast, brightness, noise, pixel sampling, and processing. A high-contrast pair can be detected below the Rayleigh criterion, and a low-contrast pair can be missed above it. Treat the criterion as a physical baseline for comparing systems. The full Rayleigh criterion derivation works through where the 1.22 comes from, the 26.5 percent dip at the limit, and how the criterion compares with the Sparrow, Abbe, and Dawes limits.

The Airy Disk as a Point Spread Function

A point spread function, or PSF, answers a simple question: if the scene contains one perfect point of light, what shape does the imaging system record? For an ideal circular aperture the answer is the Airy pattern, which makes the Airy disk the basic blur fingerprint of a diffraction-limited lens.

Knowing the PSF lets you predict how the lens treats a more complicated scene. Imagine building the scene from many tiny points. Each point is replaced by a copy of the PSF, and the copies overlap and add. In image processing, that operation is convolution:

$$I_{image}(x, y) = I_{object}(x, y) * PSF(x, y)$$

The equation says that the image is the object after every point has been spread by the PSF. Rendering and image processing call the same rule a blur kernel, though here the kernel comes directly from wave optics rather than being chosen for artistic softness. Applying that kernel across a whole image is a neighborhood operation, the kind of work a compute shader handles well.

The explorer applies that rule to repeated black-and-white detail. The upper strip is the ideal square wave, and the lower strip is what the diffraction-limited PSF records. Two handles set the experiment: the handle above the strips sets how fine the detail is, and the aperture rim on the right sets the f-number.

Widening the PSF by stopping down the aperture flattens the blurred strip toward uniform gray. Broad bars survive because the PSF only has to reach across a small fraction of a bar to smear it. Finely packed bars lose contrast first.

The curve on the right is the modulation transfer function, or MTF. It plots the fraction of contrast that survives at each spatial frequency, from zero up to a cutoff where the transfer reaches zero:

$$f_c \approx \frac{1}{\lambda N}$$

Here $f_c$ is the cutoff frequency and $N$ the f-number. Above the cutoff the ideal system transfers no contrast at all. Below it, contrast does not stay perfect; it falls gradually, which is why an image can look soft long before a feature disappears entirely.

This is the machinery behind two familiar facts. A real lens usually reaches peak sharpness at a middle aperture, where aberration blur and diffraction blur are both controlled: opening up reduces diffraction but exposes aberrations, and stopping down reduces aberrations but enlarges the diffraction spot. And more megapixels do not guarantee more detail. Tiny pixels can sample very fine structure, but if the Airy pattern is wide compared with the pixel pitch, extra pixels record the blur more finely without recovering contrast the optics never delivered. Lens tests lean on MTF for exactly that reason: they measure contrast transfer, not whether a line pair barely survives.

Real cameras and microscopes add further effects. Aberrations can make the spot asymmetric or larger, sensor pixels sample in discrete steps, and demosaicing, sharpening, motion, and focus error all leave their marks. Diffraction is the clean baseline that remains before those complications. The companion article on computational photography with HDR, burst denoising, and super-resolution picks up the software side, and convolution and filtering in images and signals covers the same blur-and-frequency relationship outside optics.

The Same Idea Across Cameras, Telescopes, and Microscopes

The units change from one field to another, but the reasoning stays stable. A telescope talks about angular separation because stars are effectively distant point sources; for the Hubble Space Telescope, the practical question is whether nearby sources can be separated, not whether the telescope magnifies enough. A microscope talks about numerical aperture and nanometer feature spacing. A camera lens talks about focal length, f-number, pixel size, and sensor-plane blur.

All three ask the same chain of questions: how large is the diffraction pattern, how much do neighboring patterns overlap, and how much contrast survives at the detail scale that matters.

That chain is more useful than any single formula:

• Diffraction sets the smallest ideal point pattern.
• The Airy disk describes the central part of that pattern for a circular aperture.
• The PSF describes how every scene point spreads into the image.
• Overlapping PSFs explain two-point resolution.
• MTF describes how much contrast survives for repeated detail.

Once those pieces fit together, the diffraction limit becomes a working model for choosing apertures, comparing objectives, reading lens charts, and building physically plausible blur in a renderer. For the ray-optics counterpart, caustics from reflection and refraction shows how curved surfaces concentrate families of rays into bright structures.

A Practical Way to Reason About Real Resolution

When a real system is on the bench, work in this order. First, estimate the diffraction scale from wavelength and aperture: use $r_1$ for spot size on the sensor and $\theta_R$ for angular separation. Second, compare that scale with the detector. A sensor cannot recover detail the optics never delivered, but poor sampling can waste detail the optics did deliver.

Third, decide whether diffraction is actually the dominant limit. At small apertures it usually is, while at large apertures aberrations, focus error, or motion often dominate instead. Fourth, judge contrast for the detail you care about. Barely splitting two ideal points is a different task from reading a low-contrast texture, detecting a faint star, or measuring a small manufactured edge.

That order keeps the topic practical: start with the clean physics, then add the messier parts of the real system.

Recap

The Airy disk is the smallest point pattern a perfect circular system can form. It exists because light diffracts through a finite aperture, not because the lens is defective. The spot shrinks with a shorter wavelength, a wider aperture, or a shorter focal length, and it grows in the opposite directions.

That single pattern explains the rest. Two nearby Airy disks overlap, which produces Rayleigh-style resolution. Many overlapping point patterns blur the image, which is the PSF view. Repeated detail loses contrast as the PSF widens, which is the MTF view. The formulas are compact, but the mental model is simple: a finite aperture turns every point into a small structured spot, and resolution is what happens when those spots overlap.