Airy disk

The Airy disk is the diffraction pattern produced in the focal plane of an ideal converging lens (or far-field of an aperture) when light from a point source passes through a circular aperture. The pattern was first described by George Biddell Airy in 1835 and consists of a bright central disk surrounded by alternating dark and bright concentric rings.

Intensity distribution. The radial intensity profile of the Airy pattern:

$$I(\theta) \;=\; I_0 \left[ \frac{2 J_1(x)}{x} \right]^2, \quad x \;=\; \frac{2\pi a}{\lambda} \sin\theta,$$

where $J_1$ is the first-order Bessel function, $a$ is the aperture radius, and $\theta$ is the angle from the optical axis.

Positions of the dark rings. The dark rings occur at the zeros of $J_1(x)$:

Ring	Zero of $J_1$	Angular position $\theta$	Linear position at focal length $f$
1st dark ring	$x = 3.832$	$\sin\theta = 1.22 \lambda / D$	$r_1 = 1.22 \lambda f / D$
2nd dark ring	$x = 7.016$	$\sin\theta = 2.23 \lambda / D$	$r_2 = 2.23 \lambda f / D$
3rd dark ring	$x = 10.173$	$\sin\theta = 3.24 \lambda / D$	$r_3 = 3.24 \lambda f / D$

The diameter of the central Airy disk (between the two first dark rings) is $d = 2.44 \lambda f / D$.

Power distribution. The fraction of total beam power enclosed within each ring:

Ring inclusion	Fraction of total power
Central disk only (within first dark ring)	83.8%
Through 1st bright ring	91.0%
Through 2nd bright ring	93.8%
Through 3rd bright ring	95.3%

This is why the central Airy disk is the conceptual "spot size" of an imaging system: it contains nearly all the power even though the rings extend to infinity.

Diffraction-limited spot size. For a focusing system with numerical aperture NA, the Airy disk diameter is:

$$d_\text{Airy} \;=\; 1.22 \, \frac{\lambda}{\text{NA}} \;=\; 1.22 \, \frac{\lambda f}{D},$$

where the second form uses $\text{NA} = D/(2f)$ for small NA (paraxial approximation). This is the absolute lower limit of focused-spot size — no aberration-free lens can produce a smaller spot.

Examples for typical systems.

System	$\lambda$	NA or $f/\#$	Airy disk diameter
Microscope objective 100×/1.4 oil	550 nm	1.4	480 nm
Microscope objective 40×/0.65	550 nm	0.65	1.0 μm
Camera lens, f/4	550 nm	0.125	5.4 μm
Telescope, 8-inch f/10	550 nm	0.05	13.4 μm
Photolithography stepper	193 nm	0.95	250 nm
Optical disc drive (Blu-ray)	405 nm	0.85	580 nm
HeNe focused by 100 mm f/2 lens	633 nm	0.25	3.1 μm
Telecom DFB into single-mode fiber	1550 nm	0.14	13.5 μm

Rayleigh resolution criterion. Two point sources are "just resolved" when the central Airy disk of one falls on the first dark ring of the other. This separation in image space is the Airy radius:

$$\delta_\text{Rayleigh} \;=\; 1.22 \, \frac{\lambda f}{D} \;=\; 1.22 \, \frac{\lambda}{2 \text{NA}}.$$

Below this separation, two point sources merge into a single elongated blob and cannot be reliably resolved.

Sparrow criterion. A slightly tighter resolution definition: two point sources are "just resolved" when the combined intensity profile no longer has a central dip between them. The Sparrow criterion gives:

$$\delta_\text{Sparrow} \;\approx\; 0.94 \, \frac{\lambda}{2 \text{NA}},$$

about 25% smaller than Rayleigh. Modern resolution standards typically use Rayleigh, but Sparrow is sometimes cited in imaging literature.

Abbe limit (microscopy). For incoherent illumination at the focal plane of a microscope:

$$d_\text{Abbe} \;=\; \frac{\lambda}{2 \text{NA}},$$

approximately 22% smaller than the Rayleigh-limited diffraction spot. The Abbe limit is the most commonly cited resolution limit in microscopy.

Beyond the Airy disk. Several techniques achieve sub-Airy-disk resolution:

• Structured illumination microscopy (SIM): extracts spatial frequency components beyond the cutoff via interference patterns; resolution ~2× better than Abbe
• Stimulated emission depletion (STED): depletes outer parts of the Airy disk by stimulated emission; effective resolution down to ~30 nm
• PALM/STORM: localizes individual fluorophores to ~10 nm precision via Gaussian fitting of their (still-diffraction-limited) PSFs
• Near-field scanning optical microscopy (NSOM): uses sub-wavelength probes, bypassing diffraction
• Photoactivated localization: similar to PALM/STORM

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 4 (Fourier optics, diffraction); Born & Wolf, Principles of Optics (7th ed.), Ch. 8 for the rigorous derivation; Goodman, Introduction to Fourier Optics (3rd ed., 2005), Ch. 6.