Past exam of the physics 2 paper of the University of Oxford / 2016 / 6

Under the Rayleigh criterion, two equally bright, mutually incoherent point sources are just resolved when the central maximum of one diffraction pattern lies at the first minimum of the other. Their intensities, rather than their electric-field amplitudes, add. For an ideal circular aperture of diameter $D$, the first dark ring of the Airy disk givesThe angle is in radians in the small-angle limit. This is a conventional resolution criterion for a specified aperture and wavelength; it is not an absolute prohibition on estimating a source separation by fitting noisy data under additional assumptions.

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.