Point Spread Function

A point is never a point

Point a telescope at a star and you are aiming at the simplest possible target: a source so far away that, geometrically, it has no size at all. Even mighty Betelgeuse — a red supergiant 700 times wider than the Sun — subtends only about 0.05 arcseconds from Earth, smaller than what most telescopes can resolve. To the optics, a star is an ideal point source, all of its light arriving as parallel rays from a single direction.

And yet no telescope ever records a star as a single, infinitely small dot. Instead the light lands as a small bright disk surrounded by a set of faint, fading rings. That recorded shape — the image of a point — is the point spread function. It is the fingerprint of the instrument: the way the optical system "spreads" the light from one ideal point across the focal plane. Every object in an astronomical image, from a single star to a galaxy, is the true scene mathematically convolved with this same blur kernel.

The culprit is diffraction. Light is a wave, and whenever a wave passes through a finite opening — the telescope's aperture — it spreads out. The sharp edge of the mirror or lens forces the wavefront to interfere with itself, and the result, for a circular aperture, is the famous Airy pattern, named for George Biddell Airy who derived it in 1835.

The Airy disk and its rings

For a perfect circular, unobstructed aperture of diameter D observing light of wavelength λ, the diffraction PSF is the squared jinc function — a bright central peak called the Airy disk, surrounded by concentric dark and bright rings. The first dark ring sits at an angular radius of

θ = 1.22 λ/D (radians)

The numbers are striking. The Airy disk alone contains roughly 84% of the total light; about 91% lies inside the first dark ring; the entire infinite series of outer rings carries less than 9% of the energy, fading rapidly. The first bright ring peaks at only about 1.7% of the central brightness. This is why a well-focused star looks like a tight dot to the eye even though, in principle, its light extends outward forever.

Increase the aperture and the whole pattern shrinks. Double D and the Airy disk halves in angular size; the PSF gets narrower and the telescope sees finer detail. This is the single deepest reason astronomers build ever-larger mirrors — not just to gather more photons, but to shrink the PSF.

Diffraction-limited PSF size (radius to first dark ring) at λ = 550 nm

Aperture	Diameter D	θ = 1.22 λ/D	Note
Human eye (pupil)	~5 mm	~28 arcsec	Why stars twinkle as points
Backyard refractor	100 mm	~1.4 arcsec	Splits close double stars
Hubble Space Telescope	2.4 m	~0.058 arcsec	Above the atmosphere
VLT / Gemini	8 m	~0.017 arcsec	Needs adaptive optics
ELT (under construction)	39 m	~0.0036 arcsec	Will dwarf Hubble's sharpness

How the PSF sets resolution

The PSF is the reason resolution is finite. Two stars sitting close together each produce their own Airy pattern. When they are far apart, you see two clean dots. Bring them together and the patterns overlap; eventually the two peaks merge into a single elongated blob, and no amount of staring or magnification can separate them.

The conventional dividing line is the Rayleigh criterion: two equal point sources are "just resolved" when the peak of one Airy pattern falls exactly on the first dark ring of the other — a separation of θ = 1.22 λ/D. At that spacing a shallow dip appears between the two peaks. (The stricter Sparrow limit, at about 0.95 λ/D, marks where the dip vanishes entirely and the pair looks single.) The deciding factor is always the same: the separation of two sources must exceed the width of the PSF for them to be told apart.

This is why the diameter of a telescope, not its magnification, sets how much detail it can show. A small telescope with high magnification merely produces a bigger, blurrier blob; a large telescope produces a smaller PSF and genuinely new detail.

When the atmosphere widens the PSF

The diffraction limit is the best a telescope can do. From the ground it almost never gets there. Earth's turbulent atmosphere is filled with pockets of air at slightly different temperatures and densities, each refracting light a little differently. As these cells drift across the aperture they scramble the incoming flat wavefront, and the once-tidy Airy pattern dances and smears into a fuzzy blob typically 0.5 to 1.5 arcseconds wide — vastly larger than the ~0.017 arcsec diffraction limit of an 8 m mirror.

This atmospheric broadening is called the seeing, and it dominates the observed PSF for any large ground-based telescope. The PSF you actually record is the convolution of every blur in the chain:

• Diffraction — the irreducible Airy core set by aperture size.
• Atmospheric seeing — the biggest contributor from the ground.
• Aberrations — coma, astigmatism, spherical aberration, defocus.
• Central obstruction — the secondary mirror pushes more light into the rings and adds diffraction spikes from its support spider.
• Tracking and guiding errors — imperfect motion smears the spot during a long exposure.
• Detector pixel size — the PSF must be sampled by at least ~2 pixels (Nyquist) or detail is lost.

What sets the PSF width in different regimes

Setting	Dominant PSF driver	Typical FWHM
Naked-eye ground	Eye optics + seeing	~60 arcsec (eye limit)
Ground, large telescope, no correction	Atmospheric seeing	~0.5–1.5 arcsec
Ground + adaptive optics	Residual + diffraction	~0.03–0.1 arcsec
Space telescope (HST/JWST)	Diffraction limit	~0.05–0.1 arcsec

Two technologies beat the seeing. Adaptive optics measures the distorted wavefront hundreds of times per second and reshapes a deformable mirror to cancel the turbulence, restoring a near-diffraction-limited PSF. And putting a telescope in space removes the atmosphere entirely — which is why Hubble and JWST reach their full theoretical sharpness.

Reading and undoing the blur

The PSF is not just a nuisance to be minimised — once you know it, it becomes a powerful tool. Because the recorded image is the true sky convolved with the PSF, knowing the PSF lets you partly reverse the operation. This is deconvolution.

Astronomers measure the PSF directly: isolated, unsaturated stars in the same frame are themselves point sources, so each one literally is a snapshot of the PSF. Software such as DAOPHOT builds an empirical PSF model from these stars; for space telescopes, simulation tools like WebbPSF model how the PSF varies across the field. With that model in hand:

• Deconvolution algorithms — Richardson–Lucy for optical images, CLEAN for radio interferometry — sharpen the image and pull apart close sources, within the limits the PSF preserved.
• PSF photometry fits the model PSF to every star in a crowded field (like a globular cluster), measuring individual brightnesses even where the stars overlap.
• PSF subtraction removes a bright star's pattern to expose faint companions buried in its glare — the core trick behind directly imaging exoplanets next to their host stars.
• Astrometry centroids the PSF to locate a star to a small fraction of a pixel, enabling milliarcsecond position measurements.

One hard rule applies: deconvolution can never recover detail finer than the diffraction limit. Information the aperture never collected is simply gone. What sharpening recovers is the structure the PSF blurred but did not destroy.

Why the PSF matters

• It defines resolution. The PSF width — usually quoted as the FWHM — is the practical measure of how sharp an image is.
• It calibrates every measurement. Photometry, astrometry and morphology all depend on an accurate PSF model.
• It diagnoses the instrument. A drifting PSF shape flags focus drift, optical misalignment or worsening seeing in real time.
• It enables exoplanet imaging. Subtracting a stable, well-known PSF is how faint planets are pulled out of starlight a million times brighter.
• It is the bridge to deconvolution. Knowing the blur is the prerequisite for undoing it.

Common misconceptions

• "More magnification means more detail." No — resolution is fixed by the PSF (aperture and seeing). Extra magnification just enlarges the blur.
• "Big stars look bigger in images." Nearly all stars are unresolved points; a brighter star's larger disk is just a wider PSF, not its true size.
• "Perfect optics give a perfect dot." Even flawless optics are diffraction-limited — the Airy disk is the smallest possible spot.
• "Deconvolution can recover any lost detail." It can only restore what the PSF preserved; nothing finer than the diffraction limit survives.
• "The PSF is the same everywhere in an image." It varies with field position, wavelength, focus and seeing — which is why it must be measured, not assumed.