= Point spread function
{title2=$\operatorname{PSF}$}
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= PSF
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A <point spread function> is an imaging system's response to an <optical point source>. Here it means the distribution of image <light intensity>: an ideal point object becomes a finite spot, often with faint rings, because of <diffraction>. An unresolved <star> and a sufficiently small bead emitting <fluorescence> are practical approximations to an <optical point source>. Their image tells us how nearby object details will overlap, and reveals blurring from <optical aberrations>, <defocus>, and, for an <optical telescope>, <astronomical seeing>.
A useful convention separates shape from throughput. At a fixed image plane, let $p(\mathbf x)\geq0$ be the <point spread function> with $\int_{\mathbb R^2}p(\mathbf x)\,d^2\mathbf x=1$. Its units are inverse <area>. A point delivering detected <radiant flux> $F$ at image position $\mathbf x_0$ produces <light intensity> $Fp(\mathbf x-\mathbf x_0)$; this <point spread function normalization> makes the <integral> over the image equal to $F$. A profile divided by its own peak is convenient for comparing widths, but is a different <point spread function normalization> and does not by itself compare total <radiant flux>.
In <incoherent imaging>, different object points have negligible mutual <spatial optical coherence> over the measurement, so their image <light intensities> add. If the response also obeys <translation invariance> over the chosen <field of view>, the expected continuous image is the <convolution>
$$
I(\mathbf x)=\tau\int_{\mathbb R^2}O(\mathbf u)p(\mathbf x-\mathbf u)\,d^2\mathbf u=\tau(O*p)(\mathbf x).
$$
Here $O$ is the object brightness distribution expressed in image-scaled coordinates, with <magnification> already included, and $\tau$ is constant transmission. For $O(\mathbf u)=F_0\delta(\mathbf u-\mathbf u_0)$, where $\delta$ is the two-dimensional <Dirac delta function>, this gives $I=\tau F_0p(\mathbf x-\mathbf u_0)$. A changing response across the <field of view> instead requires a position-dependent <integral kernel> $p(\mathbf x;\mathbf u)$; one measured <point spread function> is then only a local calibration.
In <coherent imaging>, the <superposition principle> applies to the complex <wave amplitudes>. With <amplitude spread function> $h$, the corresponding ideal model is
$$
E_{\rm image}=h*E_{\rm object},\qquad I_{\rm image}=|E_{\rm image}|^2,\qquad p=\frac{|h|^2}{\int_{\mathbb R^2}|h|^2\,d^2\mathbf x}.
$$
Thus $|h*E|^2$ generally differs from $|h|^2*|E|^2$: coherent contributions produce <wave interference> terms that depend on their relative <wave phases>. Their time average vanishes for mutually incoherent emitters, which explains the intensity <convolution>. Partial <optical coherence> requires the correlations between object points as well.<MIT 2.71 and 2.710 Optics, Spring 2014 by Nicholas X. Fang/Review for Optics>[pp. 7–8, §E(a–b): coherent field and incoherent intensity imaging]{ref}
\Image[/point-spread-two-source-overlap.png]
{title=Ideal circular-pupil point responses and the overlap of two mutually incoherent sources}
The figure is an original <diffraction> model, not measured data. All panels have the same total <radiant flux>: each member of a pair contributes half the single-source <radiant flux>. Dashed curves show the separate <light intensity> contributions; the solid curve is their sum. The <point spread functions> have tails, so even sources satisfying the <Rayleigh criterion> overlap. As separation decreases, the image changes continuously; there is no abrupt disappearance of all information at one separation.
A motivating measured example is <Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others> (2005). A $543.8\,\mathrm{nm}$ <laser>, delivered through an <optical fiber>, excited fluorescent beads on a slide through a $100\times$, <numerical aperture> $1.3$ oil-immersion objective. A moving stage scanned the bead; emitted <light> returned through the objective, an emission filter and a pinhole to a <photodiode>. The readout was <fluorescence> <light intensity> against scanned position. Under the same setup, nominal bead <diameters> $0.17$, $0.5$, and $1\,\mathrm{\mu m}$ produced progressively broader measured spots. The size comparison controls for the finite object: the measured response is the <convolution> of bead emission with the effective <point spread function>, rather than a pure point response. Inverting this <convolution> model gave a response compared with the theoretical one. The smaller bead gave a weaker <signal> relative to <noise>; uncertainty in actual bead <diameter> also affected the restored width. These are limits of the calibration. In <confocal microscopy>, excitation and pinhole detection jointly determine the effective <point spread function>; this experiment does not measure the ideal widefield <Airy disk> plotted above.<Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others>[§2 and Fig. 1, pp. 2–3: apparatus; §3.2 and Figs. 4–5, pp. 4–5: bead-size comparison and limitations]{ref}
For a new calibration, useful additional controls are to measure detector background, avoid saturating the <photodetector>, repeat the scan to check stability, and compare isolated beads at several positions in the <field of view>. Record <wavelength>, <numerical aperture>, pinhole setting, sampling interval and <optical focus>. These are practical recommendations, not extra measurements claimed for the published experiment.
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