The Airy Disk and the Diffraction Limit: Why Every Lens Blurs (and How Bragg Turned Blur into Crystal Maps)

Point the best telescope ever built at a star and you will not see a point. You will see a small bright blob surrounded by faint rings. This is not a manufacturing flaw. It is what light does because it is a wave, and it is the subject of our Airy Disk hoodie, whose slogan says it plainly: even a perfect lens blurs. This post builds on our explainer How Are Electricity and Magnetism Connected?, because light is an electromagnetic wave, and everything below follows from that.

What is the Airy disk?

When light passes through a circular opening, such as a lens or a telescope mirror, it spreads out slightly. Waves from different parts of the opening arrive at the screen with slightly different path lengths, so they add up in some directions and cancel in others. For a circular aperture the result is a bright central spot with dim rings around it. The central spot is called the Airy disk, and Nikon’s microscopy guide describes the limit of resolution as the ability “to distinguish between two closely spaced Airy disks” (Nikon MicroscopyU^[1]).

The bright pattern has a precise mathematical shape. For a perfect circular aperture of diameter D, the intensity at angle θ from the centre is

I(θ) = I₀ [2J₁(x)/x]², with x = πD sin θ / λ

Here I₀ is the intensity at the centre, λ is the wavelength of the light in metres, D is the aperture diameter in metres and J₁ is a standard mathematical function called the Bessel function of the first kind, order one. The first dark ring falls where J₁(x) first reaches zero, at x ≈ 3.8317. Dividing 3.8317 by π gives 1.22, which is where the famous number comes from.

What is the diffraction limit formula?

OpenStax’s university physics text states it directly: for a circular aperture of diameter D, “the first minimum in the diffraction pattern occurs at θ = 1.22 λ/D”, provided the aperture is large compared with the wavelength (OpenStax University Physics, §4.5^[2]). Written as a rule for resolution:

θ = 1.22 λ/D

θ is the smallest angle (in radians) between two point sources that can still be told apart. Two things follow at once. Shorter wavelengths give sharper images, and bigger apertures give sharper images. That is why astronomers build ever larger telescopes and why electron microscopes, which use waves far shorter than visible light, can see much finer detail.

What is the Rayleigh criterion?

The Rayleigh criterion, developed by Lord Rayleigh in the nineteenth century, says two images are “just resolvable when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other” (OpenStax §4.5^[2]). Our second diagram below shows two Airy patterns at exactly that spacing: the summed brightness dips by about a quarter between the peaks, just enough for the eye to see two stars instead of one.

How sharp can the Hubble Space Telescope see?

Hubble’s primary mirror is 2.4 metres across (NASA^[3]). OpenStax works the example with λ = 550 nm (green light) and D = 2.40 m:

θ = 1.22 × (550 × 10⁻⁹ m) / (2.40 m) = 2.80 × 10⁻⁷ rad

That is about 0.058 arcseconds (one radian is 206 265 arcseconds). NASA says Hubble “can distinguish astronomical objects in visible light with an angular diameter of a mere 0.05 arcsecond” (NASA^[3]), so the telescope works close to the limit physics allows. Being above the atmosphere matters: OpenStax notes that in orbit the telescope “avoids the degrading effects of atmospheric distortion on its resolution” (OpenStax §4.5^[2]), and NASA says Hubble’s resolution is “ten times better than the best typically attained by even larger, ground-based telescopes” (NASA^[3]).

Why can’t an ordinary microscope see a virus?

Microscopes face the same wall. Nikon gives the resolution of a microscope objective as r = 0.61λ/NA, where r is the smallest resolvable distance and NA is the numerical aperture, a measure of the cone of light the lens collects (Nikon MicroscopyU^[1]). With green light (λ ≈ 500 nm) and a very good oil-immersion objective (NA ≈ 1.4), r ≈ 0.61 × 500 nm / 1.4 ≈ 220 nm.

The Nobel committee tells the history: in 1873 Ernst Abbe published an equation showing how microscope resolution is limited by the wavelength of light, and for most of the 20th century scientists believed they could never see things smaller than roughly “0.2 micrometres” in an optical microscope (Nobel Prize in Chemistry 2014, popular information^[4]). The 2014 chemistry prize went to Eric Betzig, Stefan Hell and William Moerner for finding ways around that limit using fluorescent molecules. Their methods do not abolish diffraction. They control which fluorescent molecules glow, so that each one can be located more precisely than the old limit; the Nobel committee describes sparse glowing molecules as “tiny lamps with switches” (Nobel Prize in Chemistry 2014^[4]).

How did diffraction reveal the structure of crystals?

If a wave passing through a hole spreads into a pattern, a wave bouncing off a regular grid of atoms should spread into a pattern too. X-rays have wavelengths comparable to atomic spacings, which are on the order of 0.1 nm (OpenStax §4.6^[5]). In 1912 Max von Laue convinced two colleagues to scatter X-rays from crystals, and the patterns proved that X-rays are waves (OpenStax §4.6^[5]). Von Laue received the 1914 Nobel Prize in Physics “for his discovery of the diffraction of X-rays by crystals” (NobelPrize.org^[6]).

What is Bragg’s law?

William Henry Bragg and his son William Lawrence Bragg found the simple rule for when the reflected waves line up. Treat the crystal as a stack of atomic planes a distance d apart. Waves reflecting from neighbouring planes travel extra distance 2d sin θ, and they reinforce each other only when that extra distance is a whole number of wavelengths:

mλ = 2d sin θ

Here m = 1, 2, 3… is the order of the reflection, λ is the X-ray wavelength, d is the spacing between atomic planes (both in metres), and θ is the angle between the incoming beam and the planes. OpenStax lists it as the Bragg equation (OpenStax key equations^[7]). As a worked number, take X-rays of wavelength 0.100 nm on planes 0.200 nm apart: for m = 1, sin θ = 0.100/(2 × 0.200) = 0.25, so θ ≈ 14.5°. Measure θ, know λ, and you have measured the spacing between rows of atoms. That is the idea on our Diffraction hoodie: spots appear only where the rows agree.

The Nobel Prize summary credits the Braggs’ contribution as “establishing the relationship between the wavelength of the X-ray, its angle of incidence, and the distance between the atomic layers inside the crystal” (NobelPrize.org^[8]). They shared the 1915 Nobel Prize in Physics “for their services in the analysis of crystal structure by means of X-rays” (NobelPrize.org^[9]). Lawrence Bragg did his prize-winning work “when he was only 23, partly in collaboration with his father” (NobelPrize.org^[8]). OpenStax calls them “the unique father-and-son team” (OpenStax §4.6^[5]).

Is the Airy disk the same physics as the double slit?

Yes. A circular hole, two slits and a crystal lattice are all openings or obstacles that make waves interfere. Change the shape and you change the pattern: two slits give stripes, a round hole gives rings, a crystal gives a lattice of spots. Our Double Slit hoodie shows the stripes, and our post Does Observation Change a Particle? explains what happens when you send the light through one photon at a time.

FAQ

Can a better lens beat the diffraction limit?

No. Polishing removes aberrations, but θ = 1.22 λ/D applies to a perfect lens. Only a larger aperture, a shorter wavelength, or a trick that changes how the light is produced (like the fluorescence methods above) does better.

Why 1.22 and not 1?

For a slit the first dark fringe is at sin θ = λ/D. For a round opening the geometry produces the Bessel function J₁, whose first zero sits at 3.8317, and 3.8317/π ≈ 1.22.

Does Bragg’s law work for electrons?

The geometry is the same for any wave, so it applies wherever particles behave like waves. This post sticks to X-rays, the case the Braggs studied.

References

1. Nikon MicroscopyU, “Resolution” (Microscopy Basics). https://www.microscopyu.com/microscopy-basics/resolution
2. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §4.5 “Circular Apertures and Resolution”. https://openstax.org/books/university-physics-volume-3/pages/4-5-circular-apertures-and-resolution
3. NASA Science, “Hubble Space Telescope: Optics”. https://science.nasa.gov/mission/hubble/observatory/design/optics/
4. NobelPrize.org, Popular information: The Nobel Prize in Chemistry 2014 (super-resolved fluorescence microscopy). https://www.nobelprize.org/prizes/chemistry/2014/popular-information/
5. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), §4.6 “X-Ray Diffraction”. https://openstax.org/books/university-physics-volume-3/pages/4-6-x-ray-diffraction
6. NobelPrize.org, The Nobel Prize in Physics 1914 (Max von Laue). https://www.nobelprize.org/prizes/physics/1914/summary/
7. S. J. Ling, J. Sanny & W. Moebs, University Physics Volume 3 (OpenStax, Rice University), Ch. 4 Key Equations. https://openstax.org/books/university-physics-volume-3/pages/4-key-equations
8. NobelPrize.org, Lawrence Bragg – Facts (Nobel Prize in Physics 1915). https://www.nobelprize.org/prizes/physics/1915/wl-bragg/facts/
9. NobelPrize.org, The Nobel Prize in Physics 1915 (William Henry Bragg and William Lawrence Bragg). https://www.nobelprize.org/prizes/physics/1915/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.