Tool · Imaging and diffraction

Airy Disk Calculator

How large is the smallest spot a circular aperture can form at a given wavelength and f-number or numerical aperture, and how close can two points be before they merge? The calculator evaluates the Airy pattern from Bessel functions and reports the disk radius and diameter, the FWHM, the ring positions, the encircled energy, the Rayleigh, Sparrow and Abbe limits, angular resolution for a given aperture, and pixel sampling for a camera. Background: Airy disk, diffraction limit, and point spread function.

Optical system

Wavelength λ (nm)

f-number N = f / D

image-side NA = 1 / (2N) for a lens in air (sine condition)

Aperture diameter D (mm)

Focal length f (mm)

Numerical aperture NA

Immersion index n

Magnification to camera M

object-plane pattern; NA = n sin θ is limited to the immersion index

Camera pixel pitch p (µm)

Nyquist sampling of the incoherent diffraction-limited image, whose finest detail is at the cutoff 2NA/λ, needs a pitch p ≤ λ/(4NA) = λN/2 (p/M at the sample in the NA mode). The Airy radius is 1.22 λN, so this is 2.44 pixels per Airy radius; coarser pixels alias detail that the optics pass.

Energy probe radius (Airy radii)

reads the fraction of the total power inside a circle of this radius

Presets

Scalar diffraction theory for a uniformly illuminated, aberration-free circular aperture. NA = n sin θ under the sine condition, so NA = 1/(2N) for a lens in air. Above about NA 0.7 the vector treatment (Richards and Wolf 1959) elongates the spot slightly along the polarization direction; the values here are the scalar ones that the resolution criteria quote.

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Readouts

Ring	Dark-ring radius	Energy inside	Bright-ring peak

Radial intensity profile

Encircled energy

Two-point resolution

separation × Rayleigh

source 1source 2incoherent sum

The pattern

Checked against

These checks run in your browser on every load. The Bessel functions are compared with the Abramowitz and Stegun tables, the ring positions with the tabulated zeros of J₁ and J₂, the encircled energies and the Rayleigh saddle with Born and Wolf, the Sparrow limit with its published value, and the unit conversions with the textbook 2.44 λN, 0.61 λ/NA and 138/D rules.

Check	Expected	Computed	Tolerance

The table sets each value this page computes beside a published reference: Abramowitz and Stegun for the Bessel functions and their zeros, Born and Wolf for the encircled energies and the Rayleigh dip, Sparrow (1916) for the Sparrow limit, and the textbook 2.44 λN, 0.61 λ/NA and 138/D rules for the unit paths. The tolerance is the largest difference from Expected, in the same units, that still passes.

The model

A uniformly illuminated, aberration-free circular aperture focuses a plane wave to the Fraunhofer pattern

I (x) = I_{0} {[\frac{2 J_{1} (x)}{x}]}^{2}

x = \frac{2 π N A r}{λ}

where $r$ is the radius in the focal plane and $J_{1}$ the Bessel function of the first kind. The fraction of the power inside radius $r$ is

E (x) = 1 - J_{0}^{2} (x) - J_{1}^{2} (x)

The first dark ring lies at the first zero of $J_{1}$ , $x = 3.8317$ , and the intensity is half its peak at $x = 1.6163$ . For a lens in air the tool takes $N A = 1 / (2 N)$ , which gives

r_{1} = 1.2197 λ N

F W H M = 1.0290 λ N

The Rayleigh limit places one point on the other’s first dark ring (separation $r_{1}$ ); the Sparrow limit is the separation at which the dip between two equal incoherent points just vanishes, $0.9472 λ N$ ; the Abbe limit is $λ / (2 N A)$ . Nyquist sampling of the incoherent image needs a pixel pitch no larger than $λ / (4 N A)$ , about 2.44 pixels per Airy radius. The model is scalar and monochromatic. It does not include aberrations, defocus, a central obstruction (which moves energy from the core into the rings), a Gaussian or otherwise nonuniform pupil illumination, or the polarization effects that change the focal spot at high NA.

Worked example

The default preset is a camera lens at f/4 with 550 nm light and 3.76 µm pixels. Then $r_{1} = 1.2197 \times 0.55 \times 4 = 2.68 μ m$ , so the disk diameter is 5.37 µm; the FWHM is 2.26 µm, the Sparrow limit 2.08 µm and the Abbe limit 2.20 µm. The central disk holds 83.8 % of the power, and the first bright ring peaks at 1.75 % of the central intensity at 3.60 µm. The pixels give 2.68/3.76 = 0.714 pixels per Airy radius, well below 2.44, so the tool marks the image as undersampled; Nyquist sampling at this pixel pitch needs $N \geq 2 p / λ$ , which is f/13.7.

References: G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Transactions of the Cambridge Philosophical Society 5, 283–291 (1835). M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (1999), section 8.5.2. C. M. Sparrow, “On spectroscopic resolving power,” Astrophysical Journal 44, 76–86 (1916).