About the Airy Disk
When light passes through any circular aperture — a camera lens iris, a telescope objective, a microscope condenser — the wave nature of light causes it to diffract. Even a theoretically perfect optical system cannot focus a point source to a geometric point; instead, it produces a bright central disk surrounded by concentric rings. This is the Airy pattern, described by George Biddell Airy in 1835.
The intensity distribution is I(r) = I₀·[2J₁(x)/x]², where x = πDr/(λf), D is the aperture diameter, λ is the wavelength, f is the focal length, and J₁ is the first-order Bessel function of the first kind. The first dark ring (first zero of J₁) occurs at x = 3.8317, giving the Airy disk radius r₀ = 1.22λf/D.
The Rayleigh criterion defines the resolution limit: two point sources are just resolved when the centre of one Airy disk falls on the first dark ring of the other. The angular resolution is θ = 1.22λ/D. This simulation renders the 2D Airy pattern as a radially symmetric intensity image and a radial cross-section profile showing the ring structure and the Rayleigh limit.
Frequently Asked Questions
What is the Airy disk?
The Airy disk is the bright central spot of the diffraction pattern formed when light passes through a circular aperture. Named after astronomer George Biddell Airy, it results from the wave nature of light: even a perfect lens cannot focus a point source to a geometric point; instead, diffraction spreads it into a bright central disk surrounded by fainter concentric rings. The pattern's intensity follows I(r) = I₀[2J₁(πDr/λf)/(πDr/λf)]².
What is the Rayleigh criterion for resolution?
The Rayleigh criterion states that two point sources are just resolved when the central maximum of one Airy pattern coincides with the first minimum of the other. The first zero of J₁(x) occurs at x = 3.8317, giving a minimum angular separation of θ = 1.22λ/D and a minimum spatial separation at the focal plane of r₀ = 1.22λf/D.
Why does a smaller aperture produce a larger Airy disk?
The Airy disk radius is inversely proportional to aperture diameter: r₀ = 1.22λf/D. A smaller aperture means stronger diffraction — the light waves bend more around the edges of the opening, spreading energy over a wider angle. This is why stopping down a camera lens past its diffraction limit reduces sharpness even though depth of field increases.
How does wavelength affect optical resolution?
Shorter wavelengths produce smaller Airy disks: r₀ = 1.22λf/D is proportional to λ. Blue light (450 nm) gives ~30% better diffraction-limited resolution than red light (650 nm). This is why UV microscopy and extreme ultraviolet lithography at ~13 nm can resolve features 50 times finer than visible-light optics.
What is the f-number and how does it relate to resolution?
The f-number (f/#) = f/D is the ratio of focal length to aperture diameter. The Airy disk radius is approximately r₀ ≈ 1.22λ(f/#). For green light at f/8, r₀ ≈ 5.4 μm. Diffraction becomes the resolution limit at roughly the f-number where the Airy disk equals two pixel widths.
Can optical systems beat the Rayleigh diffraction limit?
Yes, with super-resolution techniques. STED microscopy uses two beams to shrink the effective PSF. PALM and STORM use single-molecule localisation. Structured illumination microscopy (SIM) doubles resolution. These techniques earned the 2014 Nobel Prize in Chemistry. In lithography, immersion optics and multiple patterning push beyond conventional limits.
What causes the rings around the Airy disk?
The rings are caused by constructive and destructive interference of waves diffracted from different parts of the circular aperture. The Airy pattern is the squared 2D Fourier transform of the circular aperture function. The first ring contains about 1.75% of the total energy, and successive rings contain decreasing fractions.
How is the Airy disk relevant to telescope resolution?
For astronomical telescopes, θ = 1.22λ/D gives the angular resolution in radians. A 100 mm aperture telescope at 550 nm resolves 1.4 arcseconds. The Hubble Space Telescope (2400 mm) resolves 0.05 arcseconds. Ground-based telescopes are typically seeing-limited rather than diffraction-limited, which is why adaptive optics and space observatories are valuable.
What is the point spread function (PSF)?
The point spread function is the image of an ideal point source through an optical system. The Airy pattern is the PSF of a diffraction-limited circular aperture. Any extended image can be computed as the convolution of the true object with the PSF. A narrower PSF means sharper images.
How is the Airy disk used in microscopy?
In optical microscopy, the Airy disk radius is r₀ = 0.61λ/NA, where NA = n·sin(θ) is the numerical aperture. Oil-immersion objectives with NA = 1.4 achieve r₀ ≈ 200 nm in green light — the classical resolution limit that super-resolution techniques aim to surpass.