Point Spread Function

Aim a perfect lens at a single distant star and you still do not get a point. Diffraction at the finite aperture forbids it: the star's light arrives as a bright central disk ringed by faint haloes, a fixed blur pattern called the point-spread function (PSF). It is the image of an ideal point source — the optical system's impulse response in exactly the sense electrical engineers mean. And because most imaging systems are linear and shift-invariant over a small patch of field, the entire image of any scene is the ideal geometric image convolved with this one blob: paint the PSF over every point of the scene and add. What sets the PSF's shape is the exit pupil and the wavefront leaving it. In the diffraction-limited case the amplitude PSF is the Fraunhofer pattern of the pupil — the Fourier transform of the aperture — and squaring it gives the intensity PSF a detector records. For a uniformly illuminated circular pupil that intensity is the Airy pattern, $[2J_1(x)/x]^2$ , whose first dark ring sits at $1.22\,\lambda/D$ in angle (or $1.22\,\lambda\cdot f/\#$ as a radius in the image plane). Fully 84% of the light lands inside that central Airy disk; the rest leaks into the rings. The PSF is the real-space partner of the transfer functions in frequency space: its Fourier transform is the optical transfer function, and the pupil that makes a tight PSF makes a broad, high MTF. Aberrations and defocus enter as phase errors across the pupil, redistributing light out of the peak into the wings and lowering the Strehl ratio — the surviving fraction of peak intensity. That makes the PSF simultaneously the deepest statement of a system's resolution and its most practical diagnostic, from telescope design to the sub-pixel localization behind super-resolution microscopy.

Key takeaways

The PSF is the intensity image of an ideal point source — the spatial impulse response of imaging; the full image is the ideal image convolved with the PSF.
Diffraction-limited PSF $= |\mathcal{F}\{P\}|^2$ : the squared Fourier transform of the pupil, evaluated in the image plane.
A clear circular pupil gives the Airy pattern, first dark ring at $1.22\,\lambda/D$ (angle) or $1.22\,\lambda\,(f/\#)$ (image radius); 83.8% of the energy is in the central disk.
The Fourier transform of the intensity PSF is the optical transfer function; a narrower PSF ⇔ a wider, higher MTF (aperture, not magnification, sets resolution).
Aberrations and defocus are pupil phase errors: they broaden the PSF and drop the Strehl ratio $S\approx e^{-(2\pi\sigma/\lambda)^2}$ ; $S\ge 0.8$ (RMS ≈ λ/14) is the diffraction-limited threshold.
Key idea: the diffraction-limited PSF is the squared Fourier transform of the pupil, so aperture geometry directly dictates the blur — and its Fourier partner is the OTF.

The core insight

The diffraction-limited PSF is $|\mathcal{F}\{P(\xi,\eta)\}|^2$ — the squared Fourier transform of the pupil function. Aperture geometry in the pupil plane fixes the image-plane blur: a wider pupil passes higher spatial frequencies and produces a narrower PSF. Its Fourier transform is the OTF.

Where this lives today

The Airy PSF sets the diffraction-limited resolution of every telescope and microscope: two points are "just resolved" by Rayleigh's criterion when one Airy peak falls on the other's first null, angular resolution $1.22\,\lambda/D$ . This is why astronomers build ever-larger apertures and microscopists chase high numerical aperture — both shrink the PSF. Atmospheric turbulence broadens the stellar PSF and lowers its Strehl ratio; adaptive optics measures the wavefront and applies a conjugate correction with a deformable mirror to restore a near-diffraction-limited PSF, with $S$ the headline performance number. Because imaging is convolution with the PSF, knowing it enables deconvolution — Wiener and Richardson–Lucy restoration in astronomy and confocal microscopy. Most strikingly, single-molecule localization microscopy (PALM, STORM) fits the centroid of an isolated PSF to a few nanometres, far below the Airy width, turning the "blur" into a ruler. In coronagraphy the bright Airy rings are apodized away to reveal faint exoplanets, trading core width for dynamic range. The frequency-space partner of all this is developed in Optical Transfer Function.

The physical picture

Cover every point of the ideal geometric image with a small glowing blob whose shape is the PSF, and the observed image is that blob smeared across the scene. A sharp system has a tight blob; a soft one a fat blob. The blob is set by the aperture — which is why a bigger telescope, not more magnification, sees finer detail. Magnification only enlarges the blur along with everything else. Aperture and PSF are Fourier partners, and the reciprocity is strict: a wide pupil supports high spatial frequencies and makes a narrow central lobe; a small pupil truncates them and makes a broad one. This is the imaging twin of time–bandwidth reciprocity in signal processing — narrow in one domain forces wide in the other. Phase errors across the pupil (defocus, coma, spherical aberration) conserve total energy but rob the central peak, pushing light into the wings; the peak flattens and the Strehl ratio falls. The punchline figure is the Airy pattern of a real f/8 lens with its first dark ring pinned at a genuine 5.37 μm; the second figure shows two such PSFs sliding together to the Rayleigh limit, where the saddle between them dips to 73.5% of the peaks — the textbook definition of "just resolved."

Figure

The Airy point-spread function of an f/8 lens — first dark ring at a real 5.37 μm, and 83.8% of the light in the central disk

Figure

Two point sources at the Rayleigh limit — the saddle between the Airy peaks dips to 73.5%

The core equations

Imaging is convolution: the image is the ideal geometric image blurred by the impulse response $h$ . For coherent light $h$ is the amplitude PSF and fields add; for incoherent light $|h|^2$ is the intensity PSF and intensities add.

g(x,y)=\iint h(x-x',y-y')\,f(x',y')\,\mathrm{d}x'\,\mathrm{d}y' \qquad (1)

where: $g$ — image distribution; $f$ — ideal geometric image; $h$ — point-spread function (impulse response); $(x,y)$ — image-plane coordinates. Reading it physically: replace every ideal point by a shifted copy of the PSF and sum. The diffraction-limited amplitude PSF is the Fraunhofer pattern of the pupil — its Fourier transform — and the detector squares it to intensity.

h(x,y)=\frac{1}{\lambda^2 d_i^2}\left|\iint P(\xi,\eta)\,e^{-i\frac{2\pi}{\lambda d_i}(x\xi+y\eta)}\,\mathrm{d}\xi\,\mathrm{d}\eta\right|^2 \qquad (2)

where: $P(\xi,\eta)$ — complex exit-pupil function (aperture amplitude and wavefront phase); $\lambda$ — wavelength; $d_i$ — exit-pupil-to-image distance; $(\xi,\eta)$ — pupil coordinates. The Fourier kernel maps pupil position to image-plane spatial frequency $x/\lambda d_i$ . For a uniform circular pupil the transform of a disc is a first-order Bessel function, giving the Airy intensity.

I(r)=I_0\left[\frac{2J_1(x)}{x}\right]^2,\qquad x=\frac{\pi D r}{\lambda d_i} \qquad (3)

where: $J_1$ — Bessel function of the first kind, order 1; $D$ — aperture diameter; $r$ — image-plane radius; $I_0$ — peak intensity (the bracket $\to 1$ as $x\to 0$ ). The first dark ring is the first zero of $J_1$ .

J_1(x)=0\ \text{at}\ x=3.8317\ \Rightarrow\ r_{\text{Airy}}=1.22\,\frac{\lambda d_i}{D}=1.22\,\lambda\,(f/\#) \qquad (4)

where: $r_{\text{Airy}}$ — radius of the first minimum; $f/\# = d_i/D$ — the working f-number. The constant is $3.8317/\pi = 1.2197 \approx 1.22$ . Aberrations enter through the pupil phase and are summarized by the Strehl ratio.

S=\frac{I_{\text{peak}}^{\text{aberrated}}}{I_{\text{peak}}^{\text{diff-limited}}}\approx e^{-(2\pi\sigma_W/\lambda)^2} \qquad (5)

where: $S$ — Strehl ratio; $\sigma_W$ — RMS wavefront error. The Maréchal criterion $S\ge 0.80$ (RMS ≈ λ/14) is the working definition of "diffraction-limited."

Building the PSF

Derivation AThe PSF is the squared Fourier transform of the pupil

The goal is to show that propagating the field of an ideal point source from the exit pupil to the image plane yields, in the diffraction-limited case, an amplitude PSF that is exactly the Fourier transform of the pupil function — the master relation that turns aperture geometry into image blur. The strategy is to write the converging spherical wave truncated by the pupil, push it through the Fresnel diffraction integral, and watch the quadratic phases cancel. Keep an eye on that cancellation: it is the reason a lens performs a clean Fourier transform rather than a Fresnel smear.

Step 1. The ideal image of a point is a converging spherical wave; at the exit pupil it is $U_p = P(\xi,\eta)\,e^{-i\frac{k}{2 d_i}(\xi^2+\eta^2)}$ . Why the quadratic phase: a wave converging to the image point carries exactly this curvature; the pupil $P$ merely stamps the aperture edge and any wavefront error onto it.

Step 2. Propagate to the image plane with the Fresnel integral, which multiplies by its own kernel $e^{+i\frac{k}{2 d_i}(\xi^2+\eta^2)}$ . Why this kernel: Fresnel diffraction over distance $d_i$ contributes precisely this quadratic pupil phase inside the integral.

Step 3. The converging-wave phase and the Fresnel kernel are equal and opposite, so they cancel, leaving $h_a(x,y)\propto\iint P\,e^{-i\frac{k}{d_i}(x\xi+y\eta)}\,\mathrm{d}\xi\,\mathrm{d}\eta = \mathcal{F}\{P\}$ . Why the cancellation is the whole point: without it the integrand keeps a curvature term and you get a Fresnel (near-field) pattern; with it, the image-plane field is a pure Fraunhofer/Fourier transform of the pupil.

Step 4. The detector records intensity, $I=|h_a|^2 = |\mathcal{F}\{P\}|^2$ ; the leftover image-plane quadratic phase drops out under the modulus. Why intensity, not amplitude, for a camera: photodetectors respond to $|E|^2$ , so the measured PSF is the squared transform. *Verified: Fourier-pair pupil↔amplitude-PSF consistent with Goodman Eq. (6-31) — verify_point-spread-function.py.*

Go deeper · Derivation B · 1 minThe Airy pattern and its 1.22 constant

Derivation BThe Airy pattern and its 1.22 constant

The goal is the Airy intensity (3) and the famous first-null radius (4) for a uniform circular pupil, the numbers that fix every diffraction-limited resolution. The strategy is to exploit circular symmetry: the 2-D Fourier transform of a rotationally symmetric pupil is a Hankel (order-0) transform, whose disc integral is a standard Bessel identity. The constant 1.22 that pops out is not magic — it is the first zero of $J_1$ divided by π.

Step 1. With $P=1$ inside radius $a=D/2$ , the transform reduces to $h_a(r)\propto\int_0^a J_0(2\pi\rho r/\lambda d_i)\,\rho\,\mathrm{d}\rho$ . Why $J_0$ appears: the angular part of a 2-D transform of a radial function integrates to the zeroth Bessel function — the Hankel transform.

Step 2. Use $\int_0^a J_0(b\rho)\rho\,\mathrm{d}\rho = (a/b)J_1(ab)$ to get $h_a\propto J_1(\pi D r/\lambda d_i)/r$ . Why this identity: it is the standard closed form for the disc Hankel integral; it delivers the order-1 Bessel function directly.

Step 3. Square and normalize with $J_1(x)/x\to\tfrac12$ as $x\to 0$ , giving $I=I_0[2J_1(x)/x]^2$ with $x=\pi D r/\lambda d_i$ . Why normalize this way: it fixes $I(0)=I_0$ so the bracket is 1 at the peak.

Step 4. The first zero of $J_1$ is at $x=3.8317$ , so $r_{\text{Airy}}=3.8317\,\lambda d_i/\pi D = 1.22\,\lambda d_i/D = 1.22\,\lambda(f/\#)$ . Why 1.22 is universal for a clear round pupil: $3.8317/\pi = 1.2197$ is a pure property of the Bessel function, independent of wavelength or aperture size. *Verified: $J_1$ first zero 3.831706, $3.8317/\pi=1.21967$ , $r_{\text{Airy}}=5.368\ \mu m$ at f/8, 550 nm; central-disk encircled energy $1-J_0^2-J_1^2 = 0.8378$ — verify_point-spread-function.py.*

Go deeper · Derivation C · 1 minFrom PSF to OTF, and why the incoherent cutoff doubles

Derivation CFrom PSF to OTF, and why the incoherent cutoff doubles

The goal is to connect this real-space blur to its frequency-space partner: the Fourier transform of the intensity PSF is the optical transfer function, and it reaches twice the coherent cutoff. The strategy uses the autocorrelation theorem on $|h_a|^2$ and a support-counting argument. This closes the loop with the companion Optical Transfer Function article — same physics, two domains.

Step 1. For incoherent light intensities add, so imaging is linear in intensity with impulse response $|h_a|^2$ ; its transform is the OTF. Why intensity-linear: incoherent sources have no fixed phase relation, so cross terms average away and only $|h_a|^2$ survives.

Step 2. By the autocorrelation theorem $\mathcal{F}\{|h_a|^2\} = H\star H$ , and since $H\propto P$ , the OTF is the normalized autocorrelation of the pupil. Why autocorrelation: the transform of a modulus-square is the self-correlation of the transform — the pupil sliding across itself.

Step 3. A pupil of width $D$ has an autocorrelation of width $2D$ , so the incoherent cutoff $f_c=D/\lambda d_i = 2\,\mathrm{NA}/\lambda$ is twice the coherent cutoff $\mathrm{NA}/\lambda$ . Why the tight PSF ⇔ high MTF: a narrow $h$ has a broad transform, so shrinking the Airy disk (bigger $D$ ) both sharpens the image and raises the cutoff. *Verified: incoherent cutoff = 2× coherent, consistent with the OTF article — verify_point-spread-function.py.*

Limiting cases

Case	Result	Meaning
Circular pupil	$I\propto[2J_1(x)/x]^2$ , first zero $r=1.22\,\lambda\,(f/\#)$	The Airy pattern; sets the Rayleigh resolution.
Rectangular pupil ( $a\times b$ )	$I\propto\mathrm{sinc}^2(ax/\lambda d_i)\,\mathrm{sinc}^2(by/\lambda d_i)$	Separable PSF; a single slit gives one sinc².
Gaussian (apodized) pupil	$P\propto e^{-\rho^2/w_p^2}\Rightarrow I\propto e^{-2r^2/w_i^2}$	No rings — sidelobes suppressed at the cost of a wider core.
Aberrated / defocused	$P=A\,e^{ikW}$ , $S\approx e^{-(2\pi\sigma_W/\lambda)^2}$	Wavefront error broadens the PSF, lowers the peak (Strehl).

Common trapThe intensity PSF is not the impulse response for incoherent imaging in amplitude

Coherent and incoherent imaging share the same amplitude PSF $h_a$ , but combine it differently, and confusing the two corrupts every resolution estimate. Under coherent light you convolve amplitudes with $h_a$ and the transfer function is the pupil itself; under incoherent light you convolve intensities with $|h_a|^2$ and the transfer function is the pupil's autocorrelation (the OTF). So the intensity PSF $|h_a|^2$ is the incoherent impulse response, but it is not the thing you convolve object amplitudes with, and its transform is the autocorrelation of the pupil, not the pupil. Mixing the amplitude and intensity pictures is how the factor-of-two cutoff gets lost.

Worked examples

1 · Airy disk of a camera lens vs its pixel (confidence builder)

An f/8 lens images at $\lambda = 550$ nm onto a sensor with 4.3 μm pixels. Is the lens diffraction-limited or pixel-limited?

Step 1. The Airy radius is $r_{\text{Airy}} = 1.22\,\lambda\,(f/\#) = 1.22\times 0.550\ \mu\mathrm{m}\times 8 = \mathbf{5.37\ \mu m}$ .

Step 2. Compare to the pixel: the Airy disk radius (5.37 μm) exceeds one pixel (4.3 μm), so the blur spot straddles more than one pixel — the system is diffraction-limited, not pixel-limited, at f/8. Stopping down further (larger f/#) only worsens it.

Step 3. As a sanity check on the constant, $3.8317/\pi = 1.2197$ , so using the exact constant gives 5.365 μm — the "1.22" rounding costs nothing at this precision.

2 · The Rayleigh resolution of a 100 mm telescope, and a Strehl budget (exam level)

A 100 mm aperture telescope with f/8 optics observes at 550 nm. Find its angular resolution and the wavefront error that keeps it diffraction-limited.

Step 1. Angular resolution (Rayleigh) $\theta = 1.22\,\lambda/D = 1.22\times 0.550\ \mu\mathrm{m}/100\ \mathrm{mm} = \mathbf{6.71\ \mu rad}$ (≈ 1.38 arcsec). Two stars closer than this merge.

Step 2. At the Rayleigh separation the saddle between the two Airy peaks sits at 73.5% of the peak — the classic "just resolved" contrast (each PSF contributes $[2J_1(x)/x]^2 = 0.368$ of its peak at the midpoint, summing to 0.735).

Step 3. For the image to stay diffraction-limited, the Maréchal criterion needs $S\ge 0.80$ . With $S = e^{-(2\pi\sigma_W/\lambda)^2}$ , setting $S = 0.82$ gives $\sigma_W = \lambda/14 \approx \mathbf{39\ nm}$ RMS — the total wavefront-error budget for optics plus atmosphere plus alignment.

Physics of the portrait

Like the transfer function, the point-spread function names a concept rather than a person — but its shape carries a famous name. The diffraction pattern of a uniformly illuminated circular aperture was first computed by George Biddell Airy, Astronomer Royal, in an 1835 paper on the diffraction of an object glass; the bright central disk ringed by faint haloes has been the "Airy pattern" ever since. The systems-theory reading — that this pattern is the impulse response and images are convolutions with it — matured a century later with Fourier optics.

Origins · 1 minpoint spread function and how this was discovered

point spread function

concept · Airy pattern (G. B. Airy, 1835) → linear-systems imaging · the impulse response of an optical system

The point-spread function is optics' impulse response, and its diffraction-limited form has a pedigree. In 1835 George Biddell Airy (1801–1892), then Astronomer Royal, worked out the intensity pattern a perfect circular telescope aperture forms from a star — the central disk plus rings now called the Airy pattern, with its first dark ring at $1.22\,\lambda/D$ . For a century this was a resolution formula; the deeper reframing came with linear-systems optics, which recognized the Airy pattern as the impulse response of the imaging system, so that any image is the object convolved with the PSF and the PSF's Fourier transform is the optical transfer function. That view turned the PSF into a working tool: its width sets telescope and microscope resolution, its peak (the Strehl ratio) grades adaptive-optics correction, and fitting its centroid to a fraction of its width underlies the super-resolution localization microscopy that won the 2014 Nobel Prize in Chemistry.

History

1835

— George Biddell Airy computes the diffraction pattern of a circular aperture (the Airy disk and rings)

1873

— Ernst Abbe links aperture, wavelength and resolution in the microscope

1879

— Lord Rayleigh states the two-point resolution criterion using the Airy first null

1934

— Frits Zernike's phase-contrast method (Nobel 1953) manipulates the pupil-plane spectrum of the PSF

2014

— PSF-centroid localization (PALM/STORM) earns the Nobel Prize in Chemistry for super-resolution microscopy

Going further

Texts. Goodman, Introduction to Fourier Optics, 3rd ed., ch. 6 · Hecht, Optics, 5th ed., ch. 10–11 · Born & Wolf, Principles of Optics, 7th ed., ch. 8–9 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 4.

Provenance (base-corpus migration). Standard results, re-verified from scratch: the pupil→PSF Fourier relation via Fresnel propagation (Goodman ch. 6); the Bessel/Airy derivation with first-zero constant $3.8317/\pi = 1.2197$ (Born & Wolf ch. 8); central-disk encircled energy 0.838; first-sidelobe 1.75%; Rayleigh two-point saddle 0.735; Strehl at λ/14 giving $S = 0.82$ . No base error found — every value reproduced by verify_point-spread-function.py (13 checks).

Builds on: Fraunhofer Diffraction · Circular Aperture Diffraction and the Airy Pattern · Fresnel–Kirchhoff Diffraction Integral. Leads to: Rayleigh Resolution Criterion · Optical Transfer Function · Spatial Filtering and the 4f System.

What exactly is the difference between the PSF and the Airy pattern?

The PSF is the general concept — the image an optical system forms of an ideal point, whatever the aperture and aberrations. The Airy pattern is the specific PSF of a perfect (diffraction-limited) system with a clear circular aperture: the $[2J_1(x)/x]^2$ disk-and-rings. Change the aperture (square, annular, apodized) or add aberrations and the PSF changes shape, but it is still the PSF. So every Airy pattern is a PSF; not every PSF is Airy.

Does a bigger telescope give a brighter or a sharper image of a star?

Both, but for different reasons. Brighter because it collects more light (∝ area ∝ D²). Sharper because the PSF narrows as $1.22\,\lambda/D$ , so the star's blur shrinks and closer pairs become resolvable. Magnification does neither on its own — it enlarges the blur along with the image. Resolution is set by the aperture diameter (or numerical aperture), full stop.

Why is 84% of the light in the central disk and not all of it?

Diffraction spreads a fraction of the energy into the rings. Integrating the Airy intensity out to the first dark ring gives $1 - J_0^2(z_1) - J_1^2(z_1) = 0.838$ at the first zero $z_1 = 3.8317$ — so 83.8% is in the central disk and about 16% leaks into the surrounding rings, most of it into the first bright ring (itself only 1.75% of the peak, but spread over a larger area). Apodizing the pupil can push more energy central at the cost of a wider core.

How can super-resolution microscopy beat the Airy limit if the PSF is fixed?

It does not sharpen the PSF; it locates it. If only one emitter within a diffraction spot is on at a time (as in PALM/STORM), its image is a lone Airy blob whose centroid you can fit to a small fraction of its width — a few nanometres — because centroid precision scales as (PSF width)/√(photons), not the width itself. Structured-illumination and STED use other tricks (spectrum folding, stimulated depletion), but localization microscopy simply turns the known blur into a ruler.

What is the Strehl ratio, and why is 0.8 the magic number?

The Strehl ratio is the peak intensity of the real (aberrated) PSF divided by the peak of the perfect PSF — how much of the central spike survives. Maréchal showed $S\approx e^{-(2\pi\sigma_W/\lambda)^2}$ for small RMS wavefront error $\sigma_W$ . Setting $S = 0.80$ gives $\sigma_W\approx\lambda/14$ , and by long convention a system with $S\ge 0.8$ is called "diffraction-limited" — its image is dominated by diffraction, not aberration. It is a peak metric, so it is more sensitive than a mere blur width.

Does defocus just make the PSF bigger, or does it change its shape?

It changes the shape. Defocus is a quadratic phase across the pupil; the PSF does not merely scale up — it develops a flatter, ring-filled profile and eventually a doughnut, and its Fourier transform (the OTF) dips through zero and goes negative, producing contrast reversal ("spurious resolution"). So a defocused system is not a blurrier version of a sharp one; it redistributes energy in a structured way that can even flip the polarity of fine detail.