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Optics Simulator

Airy Disk Simulator — Diffraction Limit and Rayleigh Resolution

Compute the diffraction-limited angle 1.22 lambda/D, focal-plane Airy radius 1.22 lambda F#, resolution at distance R and central-disk energy fraction in real time from wavelength lambda, aperture D, F-number and distance R. A 2D diffraction pattern and a 1D intensity profile make the resolution limit of telescopes and microscopes tangible.

Parameters
Wavelength lambda
nm
Aperture D
mm
F-number F#
Distance R
m

Defaults are lambda = 550 nm (green visible light), D = 100 mm (small telescope), F# = 8 (typical photographic lens) and R = 1000 m. Larger D shrinks the diffraction angle and lets the system resolve finer detail at long range.

Results
Angular resolution theta_Airy
Focal-plane Airy radius
Resolution at distance R
Central-disk energy fraction
2D Airy diffraction pattern

Bright central disk = Airy disk / dark concentric circles = first and second dark rings (zero intensity) / faint outer rings = secondary lobes. About 83.8% of the total light sits inside the central disk. Colour follows the visible-light approximation of lambda.

1D intensity profile [2 J_1(x)/x]^2

X = radius from centre r [microns] (focal-plane scaling) / Y = relative intensity I/I_0 / blue curve = Airy function [2 J_1(x)/x]^2 / red dashed lines = first dark ring (Airy radius) / yellow band = central disk region.

Theory & Key Formulas

The Fraunhofer diffraction pattern of monochromatic light of wavelength $\lambda$ passing through a circular aperture of diameter $D$ is given in terms of the Bessel function of the first kind, $J_1$.

Airy intensity pattern (with $x = (\pi D/\lambda)\sin\theta$):

$$I(\theta) = I_0\left[\frac{2\,J_1(x)}{x}\right]^2$$

Angular radius of the first dark ring (the first zero $x = 3.8317$ of $J_1$):

$$\theta_\mathrm{Airy} = 1.22\,\frac{\lambda}{D}$$

For focal length $f$, F-number $F\#=f/D$ and distance $R$ the Airy radius becomes:

$$r_\mathrm{focal} = 1.22\,\lambda\,F\#,\qquad r_R = \theta_\mathrm{Airy}\,R$$

The Rayleigh criterion fixes the smallest resolvable angular separation at $\theta_\mathrm{Airy}$. About $83.8\%$ of the total light energy is concentrated inside the central Airy disk (the rest leaks into the secondary rings), and that fraction also caps the Strehl ratio of the optical system.

What the Airy Disk Simulator does

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When I look at a star through a lens, why doesn't it focus to a perfect point? Isn't a lens supposed to do exactly that?
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Great question — that is the diffraction limit. Light is a wave, so the moment it passes through any aperture of finite diameter D it spreads out. Even a perfect lens cannot focus it to a point: the result is the Airy disk, a bright central spot surrounded by faint dark rings and side lobes. The angular radius of the central disk is theta_Airy = 1.22 lambda/D, which gives 6.71 microradians for the defaults lambda = 550 nm and D = 100 mm shown in this tool. That is the diffraction limit.
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In concrete terms, what kind of detail can a telescope resolve? Does that explain why bigger mirrors are better?
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Exactly. A 100 mm refractor delivers about 6.71 microradians, or 1.4 arcsec — barely able to separate two points 6.71 mm apart at one kilometre, which is what the resolution at distance R card prints. The Hubble Space Telescope (D = 2.4 m) reaches 0.05 arcsec; James Webb (D = 6.5 m, infrared) about 0.1 arcsec. Ground-based scopes are usually limited to about 1 arcsec by atmospheric seeing, but adaptive optics push them close to the diffraction limit. Sweep D from 1 to 1000 mm here and watch the resolution improve linearly.
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F# is the f-stop number written on a camera lens, right? Like F2.8 or F8. How does that connect to resolution?
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Yes, exactly. F# = f/D, and the focal-plane Airy radius is r = 1.22 lambda F#. With lambda = 550 nm and F# = 8 you get 5.37 microns. Modern full-frame DSLR pixels are 4 to 6 microns, so F# = 8 is already on the edge. Above F# = 11 the diffraction blur exceeds one pixel and stopping down further actually reduces sharpness. Drag the F# slider in this tool and watch the focal-plane Airy radius scale linearly — it is the physical reason photographers say "do not stop down too far".
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The 84% central-disk energy fraction does not change when I move any slider. Is that meaningful?
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Sharp eye. The exact value is 83.78% — a geometric constant of the circular aperture. It comes from the integral of [2 J_1(x)/x]^2 from 0 to 3.8317 divided by the full integral. The remaining 16.2% is spread over the secondary rings. That number also caps the Strehl ratio (peak intensity divided by the diffraction-limited peak), so even a perfect aberration-free system tops out at 83.8% energy in the central disk. To redistribute light beyond it you have to reshape the aperture, which is the trick coronagraphs and apodised pupils use.
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What is the difference between the Rayleigh and Sparrow criteria? My textbook just mentioned them.
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Both define when two points become resolvable, but with different yardsticks. Rayleigh places the centre of one Airy disk on the first dark ring of the other (theta = 1.22 lambda/D); the combined profile shows a 26% dip. Sparrow is the angle at which the dip just disappears and the profile becomes flat (theta about 0.95 lambda/D), which is more aggressive. Dawes is an empirical 19th-century telescope rule (theta about 1.02 lambda/D). This tool uses Rayleigh; modern CCD work and visual estimates pick whichever fits the contrast budget.

Frequently Asked Questions

The Airy disk is the bright central spot of the diffraction pattern produced when light passes through a circular aperture (a lens, stop or telescope mirror). It is named after the 19th-century British astronomer George Biddell Airy, who derived the analytical formula. The intensity follows I(theta) = I_0 [2 J_1(x)/x]^2 with x = (pi D/lambda) sin(theta); the first dark ring sits at x = 3.8317 and defines the Airy radius. About 83.8% of the total light energy falls inside the central disk and the rest spreads into the concentric secondary maxima. Move lambda and D in this tool to see how the pattern scales as lambda/D.
The Rayleigh criterion defines the smallest angular separation at which two point sources can be told apart as the angle at which the centre of one Airy disk lands on the first dark ring of the other. Lord Rayleigh proposed it in the late 19th century and the limit is theta_min = 1.22 lambda/D, which is the diffraction limit of any circular aperture. For lambda = 550 nm and D = 100 mm this gives 6.71 microradians (about 1.4 arcsec); two points 6.71 mm apart at 1 km are barely resolved, exactly the value the tool prints in the resolution at distance R card. Sharper criteria such as Sparrow and Dawes are used when contrast and noise matter.
The F-number F# = f/D combines focal length f and aperture diameter D and determines the focal-plane Airy radius r_Airy = 1.22 lambda F#. The derivation is one line: r = theta f = 1.22 (lambda/D) f = 1.22 lambda F#. With lambda = 550 nm and F# = 8 the radius is 5.37 microns. Once a digital sensor pixel pitch drops below about 4 microns, F-numbers above 11 push the diffraction blur past one pixel and the photograph is diffraction-limited. Sweep F# from 1 to 32 in this tool and watch the focal-plane Airy radius vary linearly.
Integrating [2 J_1(x)/x]^2 from x = 0 to the first zero x = 3.8317 (the Airy radius) and dividing by the integral over all of x yields 0.838, so 83.8% of the diffracted energy is concentrated in the central Airy disk and 16.2% spills into the outer rings. This is purely a geometric constant of a circular aperture and does not depend on lambda, D, F# or R. The same number sets the upper limit of the Strehl ratio (peak intensity divided by the diffraction-limited peak) and is why apodisation, which reshapes the aperture transmission, is required to redistribute light beyond it.

Real-World Applications

Astronomical telescopes: Increasing the aperture D is the most direct way to improve angular resolution. Subaru (D = 8.2 m) has a visible-light diffraction limit of about 0.014 arcsec, but ground-based images are usually limited to 0.5 to 1 arcsec by atmospheric seeing; adaptive optics correct the wavefront in real time to recover the diffraction limit. Hubble (D = 2.4 m) sits above the atmosphere and reaches 0.05 arcsec; James Webb (D = 6.5 m, infrared 1 to 28 microns) reaches about 0.1 arcsec. Set D in this tool from 1 m to 10 m to feel the theoretical limits of these large telescopes.

Optical microscopy: Microscopes usually express the limit through the numerical aperture NA = n sin(alpha) (which is equivalent to F#); the Abbe resolution d = 0.61 lambda/NA is the same Rayleigh criterion in disguise. With an oil-immersion 100x objective (NA = 1.4) and lambda = 520 nm the classical limit is d = 226 nm. Beating it requires super-resolution techniques (STED, PALM, STORM) that exploit different physical principles (single-molecule blinking, selective excitation) to reach 10 to 50 nm. Putting a tiny aperture and a short wavelength in this tool reproduces the microscopy regime.

Semiconductor lithography: An EUV scanner (lambda = 13.5 nm, NA = 0.33 to 0.55) has a theoretical minimum line width lambda/(2 NA) of about 13 nm; the K1 process factor (0.3 to 0.4) shrinks it further, but circular-aperture diffraction sets the physical floor of chip density. ASML's High-NA EUV machines (NA = 0.55) target 8 nm nodes and below, a direct industrial application of the Airy formula. Combine a small lambda and a large D in this tool to estimate the line-width floor.

Camera apertures: The familiar fact that "stopping a digital camera down past F11 actually reduces sharpness" is exactly the Airy disk exceeding the pixel pitch. A full-frame sensor with 6 micron pixels at lambda = 550 nm and F# = 11 produces a 7.4 micron Airy radius — larger than one pixel — so diffraction dominates resolution. Smartphone cameras with 1 micron pixels reach the limit at F2.0. Drag the F# slider here to feel why the "sweet spot" of any lens is dictated by the sensor it sits in front of.

Common Misconceptions and Pitfalls

The most common misconception is that "a high-quality lens beats the diffraction limit". Diffraction is set by the wave nature of light, not by the optical quality of the lens; even a perfect aberration-free design cannot beat theta_Airy = 1.22 lambda/D. "Diffraction-limited" is the highest possible quality grade and beyond it lies impossibility — only shorter wavelengths (UV, X-ray), larger apertures, or super-resolution methods (spatial modulation of the excitation) can break through. Drop lambda in this tool to 200 nm and watch the resolution improve by a factor of 2.75.

Next is the belief that "the F-number alone sets the resolution". The focal-plane Airy radius is indeed governed only by 1.22 lambda F#, but the angular resolution that lets you tell distant objects apart depends on theta = 1.22 lambda/D, which is independent of F# and depends only on D. Two F# = 8 lenses with D = 10 mm and D = 100 mm give the same 5.37 micron focal-plane blur, yet the larger aperture has ten times the angular resolution. The four sliders in this tool act independently for that reason.

Finally, people often assume that "only circular apertures produce Airy disks". Square apertures (some camera mirror frames) produce sinc^2 fringes and hexagonal apertures (most iris diaphragms) produce six-arm star spikes. The "diffraction spikes" seen on bright stars in Hubble images come from the four-arm secondary-mirror spider. The Airy disk is the special, circularly symmetric case. This tool covers only circular apertures, but in practice the aperture shape directly drives the diffraction pattern, so keep that in mind for non-round optics.