Point spread function (PSF)

The point spread function (PSF) of an optical system is the image it forms of an ideal point source. The PSF fully characterizes the system's spatial response to incoherent illumination: any extended object's image is the convolution of the object's true intensity distribution with the PSF.

Definition. For a 2D image system with object intensity distribution $O(x', y')$ and image-plane PSF $h(x, y)$:

$$I(x, y) \;=\; \int O(x', y') \, h(x - x'/M, y - y'/M) \, dx' \, dy',$$

where $M$ is the magnification. The image is a smoothed version of the object — high spatial frequencies above the PSF's cutoff are lost.

Ideal diffraction-limited PSF. For a perfect (aberration-free) lens with circular aperture, the PSF is the Airy pattern:

$$h(\rho) \;=\; \left[ \frac{2 J_1(\pi \rho / r_A)}{\pi \rho / r_A} \right]^2,$$

where $r_A = 1.22 \lambda / (2 \text{NA})$ is the Airy radius. This is the smallest PSF physically achievable for a given wavelength and numerical aperture.

Real-world PSFs. Real optical systems have PSFs larger than the diffraction-limited ideal because of aberrations and defocus. The total PSF combines:

• Diffraction-limited PSF: from the finite aperture
• Aberration-broadened PSF: from spherical, chromatic, coma, astigmatism aberrations
• Defocus-broadened PSF: from the object being off the focal plane
• Detector-broadened PSF: pixels of finite size contribute spatial averaging
• Atmospheric-blurred PSF (for telescopes): turbulence-induced wavefront distortions

The total PSF is the convolution of all these factors. Practical PSFs are typically 1.5 – 5× larger than the diffraction limit.

Three principal characterizations.

1. Spatial PSF: $h(x, y)$, the 2D point image
2. Optical transfer function (OTF): the Fourier transform of the PSF; characterizes how spatial frequencies are transmitted by the system
3. Modulation transfer function (MTF): the magnitude of the OTF; characterizes contrast vs spatial frequency

The MTF is the most commonly used metric in lens specifications. A diffraction-limited lens has MTF approaching unity at low spatial frequencies and falling to zero at the cutoff frequency:

$$f_\text{cutoff} \;=\; \frac{2 \, \text{NA}}{\lambda} \;\;\text{(incoherent)} \;\;\text{or}\;\; \frac{\text{NA}}{\lambda} \;\;\text{(coherent)}.$$

3D PSF. For thick samples or extended objects, the PSF is 3D and includes the depth dimension. For an ideal lens, the 3D PSF has:

• Tight central peak at the focal plane
• Conical "cigar shape" along the optical axis with FWHM $\sim 2\lambda n / \text{NA}^2$
• Oscillating amplitude rings characteristic of out-of-focus diffraction

This 3D structure is what microscope manufacturers and lithography stepper designers spend enormous effort to characterize and engineer.

Measuring PSF.

• Direct measurement: image a single point source (sub-resolution-sized aperture or fluorescent bead); the image is the PSF
• Edge spread function: image a sharp edge and differentiate; the derivative is the line-spread function, whose 2D Hankel transform gives the PSF
• Star imaging in astronomy: stars are effectively point sources; their image is the telescope+atmosphere PSF
• Calculated from optical prescription: ray-trace through the design; aberrations can be computed and the PSF predicted before fabrication

Standard PSF metrics.

Metric	Definition
Strehl ratio	Peak intensity of real PSF / peak intensity of ideal diffraction-limited PSF; 1.0 = diffraction-limited, $< 0.8$ = significantly aberrated
FWHM	Full-width at half-maximum of the central peak
Encircled energy at radius $r$	Fraction of PSF integrated power within radius $r$
80% encircled energy diameter	Diameter containing 80% of PSF energy (common engineering specification)

PSF engineering.

• Apodization: smoothing the aperture transmission reduces sidelobes but broadens the central peak
• Phase plates: engineered PSF shapes (e.g., elongated, bottle-beam, donut) for specialized applications
• Computational deconvolution: post-acquisition image processing removes the PSF blurring; works well when PSF is well-characterized and noise is low
• Adaptive optics: real-time wavefront correction shapes the PSF closer to the diffraction limit

Why PSF dominates imaging system performance. Any imaging system's ultimate quality is determined by its PSF. Diffraction-limited optics is the gold standard — for many applications (lithography, telescope, microscopy), achieving a diffraction-limited PSF is the primary engineering goal.

PSF in spectroscopy and astronomy. In point-spread-corrected analyses:

• Spectroscopic PSF: the wavelength-spread function of a spectrograph; sets the spectral resolution
• Astronomical PSF: combined telescope + atmosphere PSF; varies with time and pointing
• Coronagraphic PSF: special phase masks engineer the PSF to suppress on-axis light (for exoplanet imaging)

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 4 (Fourier optics, imaging); Goodman, Introduction to Fourier Optics (3rd ed., 2005), Ch. 6; Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 9 (aberrations and PSF).