Point spread function: Difference between revisions

Revision as of 19:06, 17 December 2021

Revision as of 12:39, 23 February 2022

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[[Image:PSF.svg|Truncation of Spherical Wave by Lens|right|thumb|400px]]

The quadratic [[lens (optics)|lens]] intercepts a ''portion'' of this spherical wave, and refocuses it onto a blurred point in the image plane. For a single [[lens (optics)|lens]], an on-axis point source in the object plane produces an [[Airy disc]] PSF in the image plane. It can be shown (see [[Fourier optics]], [[Huygens–Fresnel principle]], [[Fraunhofer diffraction]]) that the field radiated by a planar object (or, by reciprocity, the field converging onto a planar image) is related to its corresponding source (or image) plane distribution via a [[Fourier transform]] (FT) relation. In addition, a uniform function over a circular area (in one FT ___domain) corresponds to the [[Airy function#circular Airy function|Airy function]], {{nowrap|''J''₁(''x'')/''x''}} in the other FT ___domain, where {{nowrap|''J''₁(''x'')}} is the first-order [[Bessel function]] of the first kind. That is, a uniformly-illuminated circular aperture that passes a converging uniform spherical wave yields an [[Airy functiondisk]] image at the focal plane. A graph of a sample 2D [[Airy functiondisk]] is shown in the adjoining figure.

[[Image:Airy-3d.svg|[[Airy function#circular Airy function|Airy functiondisk]]|right|thumb|300px]]

Therefore, the converging (''partial'') spherical wave shown in the figure above produces an [[Airy disc]] in the image plane. The argument of the Airy function {{nowrap|''J''₁(''x'')/''x''}} is important, because this determines the ''scaling'' of the Airy disc (in other words, how big the disc is in the image plane). If Θ_max is the maximum angle that the converging waves make with the lens axis, ''r'' is radial distance in the image plane, and [[wavenumber]] ''k'' = 2π/λ where λ = wavelength, then the argument of the Airy function is: {{nowrap|kr tan(Θ_max)}}. If Θ_max is small (only a small portion of the converging spherical wave is available to form the image), then radial distance, r, has to be very large before the total argument of the Airy function moves away from the central spot. In other words, if Θ_max is small, the Airy disc is large (which is just another statement of Heisenberg's [[uncertainty principle]] for Fourier Transform pairs, namely that small extent in one ___domain corresponds to wide extent in the other ___domain, and the two are related via the ''[[space-bandwidth product]]''). By virtue of this, high [[magnification]] systems, which typically have small values of Θ_max (by the [[Abbe sine condition]]), can have more blur in the image, owing to the broader PSF. The size of the PSF is proportional to the [[magnification]], so that the blur is no worse in a relative sense, but it is definitely worse in an absolute sense.

The figure above illustrates the truncation of the incident spherical wave by the lens. In order to measure the point spread function — or impulse response function — of the lens, a perfect point source that radiates a perfect spherical wave in all directions of space is not needed. This is because the lens has only a finite (angular) bandwidth, or finite intercept angle. Therefore, any angular bandwidth contained in the source, which extends past the edge angle of the lens (i.e., lies outside the bandwidth of the system), is essentially wasted source bandwidth because the lens can't intercept it in order to process it. As a result, a perfect point source is not required in order to measure a perfect point spread function. All we need is a light source which has at least as much angular bandwidth as the lens being tested (and of course, is uniform over that angular sector). In other words, we only require a point source which is produced by a convergent (uniform) spherical wave whose half angle is greater than the edge angle of the lens.