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Coauthor of the original STORM imaging study. The method reconstructed positions from stochastic fluorescent events, connecting localization of individual emitters with images beyond the ordinary diffraction limit.^[ref]

An optical telescope collects light from distant objects and forms an image with an optical lens, a mirror, or a combination of both. The aperture controls both collected radiant power and the ideal diffraction limit; the focal length sets how an incoming angular separation becomes a physical separation in the focal plane.

For small angles, stars separated by angle $\theta$ form images separated by approximately $f\theta$ at focal length $f$. A larger magnification can make this separation easier to inspect, but cannot undo overlapping point spread functions. Atmospheric turbulence, imperfect focus and tracking motion can broaden them further. A detector at the focus records the image; an eyepiece instead presents it to the eye.

The detector is intended for arcsecond-scale astronomical seeing, rather than the telescope's diffraction limit. A circular aperture of this area has diameter $1.13\mathrm{m}$, giving a Rayleigh criterion angle about $0.11$ arcsec at the observing wavelength. One-arcsecond pixels conventionally sample an astronomical seeing profile of roughly two arcseconds across with two samples; a one-arcsecond profile would be undersampled. The pixel scale alone does not determine an exact astronomical seeing width.

Use the supplied surface brightness as radiative flux integrated over the photometric passband: its units contain no inverse hertz. Treat $h\nu$ as the representative photon energy, and assume unit throughput and quantum efficiency. A genuinely frequency-specific brightness would require a bandwidth. With angles measured consistently in arcseconds, an annulus has area $2\pi rdr$ square arcseconds, soThis is flux per collecting area; collected energy is $AtF$. There is no extra conversion to steradians when both brightness and angular area use square arcseconds.

For Poisson photon counting, a pixel with expected count $N$ has energy $e=Nh\nu$ and standard deviation ${\sigma }_e=h\nu \sqrt{N}$. Thus the local four-sigma rule gives $N=4\sqrt{N}$, or 16 photons at the boundary. With pixel area $\Delta \Omega =1{\mathrm{arcsec}}^2$,Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec andIntegrating to the model's outer radius $R=1/k=20$ arcsec instead givesThe pixel cut retains 51.6% of this total. It is an imposed isophotal photometry rule, not a fundamental limit: averaging annuli or fitting a justified profile can recover signal below the per-pixel cut.

In practice, sky background photons add shot noise even after subtraction of the mean sky. read noise, dark current, imperfect flat-field correction, throughput calibration and atmospheric extinction also matter. At large radius the dominant uncertainty can be a tiny error in the sky level, scattered light or the wings of the point spread function, rather than source photon statistics. Mask foreground stars and background objects, compare blank-sky regions and dithered exposures, and check profile recovery by inserting artificial extended sources. astronomical seeing and finite pixels convolve the intrinsic profile; model that response when extrapolating the total.

The luminosity convergence of an outer surface-brightness profile follows directly: $I\propto r^{-p}$ contributes $dF\propto r^{1-p}dr$, hence $F(<R)\propto R^{2-p}$ at large radius for $p<2$. For either shallow index considered, ever larger annuli dominate: extending the law to infinity gives an unphysical divergent luminosity. Such a law must eventually steepen or truncate. Over a finite range, however, substantial light can lie below a surface-brightness selection threshold and be mistaken for sky. This biases galaxy luminosities, the luminosity function and the inferred diffuse light budget; distant diffuse galaxies become especially incomplete because bolometric cosmological surface-brightness dimming scales as $(1+z{)}^{-4}$. An assumed finite-aperture total is therefore a model-dependent quantity.

Features only a few tens of nanometres wide lie below the ordinary visible-light diffraction limit, approximately $\lambda /(2\mathrm{NA})$. The microscope cannot resolve individual rods, and unpolarized illumination does not directly reveal the orientation-dependent optical response. Scanning electron microscopy and atomic force microscopy can resolve the surface pattern. Transmission electron microscopy is another possible choice if a suitably thin specimen can be prepared.

With $D=5\mathrm{mm}$, the circular-aperture diffraction limit is ${\theta }_R=244\lambda$ when $\lambda$ is in metres. No wavelength is specified, so a unique numerical diffraction angle cannot be deduced. Taking representative visible light at $\lambda =550\mathrm{nm}$ givesFor a roughly square receptor arrangement with surface density $n=10^5{\mathrm{mm}}^{-2}$, the characteristic receptor pitch is $s\simeq n^{-1/2}=3.16\mathrm{micrometres}$. With the retina $f_{\mathrm{eye}}=17\mathrm{mm}$ behind the lens, that pitch subtendsThis is an order-of-magnitude estimate of the point-separation scale from retinal sampling. A periodic intensity pattern needs at least two samples per period under the Nyquist–Shannon sampling theorem, giving a corresponding period angle of about $3.72\times 10^{-4}\mathrm{rad}$ or $1.28\mathrm{arcmin}$. A different receptor packing changes the pitch by an order-one geometric factor; density alone does not specify a full sampling lattice or a contrast threshold.

At a $5\mathrm{mm}$ pupil and the chosen visible wavelength, the optical diffraction limit is somewhat finer than the receptor pitch. The supplied model is therefore not purely diffraction limited: retinal sampling is comparable to, and slightly more restrictive than, ideal aperture diffraction. Real optical aberrations and neural processing can further reduce resolution. Different wavelengths or a smaller pupil can change which effect dominates.

The photon relation gives $\lambda =hc/E=2.4797\times 10^{-12}\mathrm{m}$. The diffraction-limited angular scale of an aperture of diameter $D$ is thereforeFor a uniformly illuminated circular aperture, the Rayleigh criterion instead gives $1.22\lambda /D=3.03\times 10^{-11}\mathrm{rad}$. Both express the same requested estimate. This is an ideal diffraction limit, conditional on the effective diameter behaving as a coherent imaging aperture; photon energy and diameter alone do not specify the performance of a practical gamma-ray telescope.

Stimulated emission depletion microscopy narrows the region from which detected fluorescence is allowed to originate. After excitation, a second beam drives stimulated emission in the surrounding region; a central minimum of that depletion beam leaves a smaller region able to emit the collected fluorescence.

Scanning this region builds an image with detail beyond an ordinary excitation-only diffraction limit. The depletion light must be rejected from the detection channel, and the central minimum must be well aligned with the excitation focus. Increasing depletion intensity can narrow the effective emitting region, but bleaching, background and specimen changes limit the useful improvement.^[ref]

Super-resolution microscopy recovers spatial detail beyond the conventional diffraction limit by acquiring information beyond a single ordinary image. Stimulated emission depletion microscopy confines detectable fluorescence with a depletion beam; photoactivated localization microscopy separates emitters in time and estimates each emitter's position before combining the estimates.

The distinction between finding one isolated emitter and resolving many adjacent emitters matters. An isolated point spread function can be localized more precisely than its width if enough photons are collected, but overlapping unknown emitters cannot simply be treated as isolated ones. Label density, photon shot noise and specimen motion therefore constrain the final optical resolution, even when the reconstructed image uses very small pixels.