HKDSE • Wave optics / Atomic World / Medical Physics

Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one Airy disk sits on the first minimum of the other. For a circular aperture, $\theta \approx 1.22\,\lambda/d$. Drag a source, then watch $\lambda$ and $d$ change the blur.

Parameters

Instrument

Drag a source in 3D, or snap to the Rayleigh limit.

Results

JUST RESOLVED
$\theta_{\min} = 1.22\,\lambda/d$ 1.74″
Actual $\theta$ 1.74″
$\theta / \theta_{\min}$ 1.00
Central dip 26%

HKDSE formula

$$ \theta \approx 1.22\,\frac{\lambda}{d} $$
$\theta$: minimum angular separation (rad)
$\lambda$: wavelength
$d$: aperture / lens diameter

Just resolved: the central maximum of one diffraction pattern coincides with the first minimum of the other.

3D optical bench
Drag source • rotate • pinch zoom
Source A Source B Aperture $d$ Image plane
Focal-plane Airy disks

Intensity along the line of centres

Physics note

A circular aperture diffracts each point source into an Airy disk. Larger $d$ or shorter $\lambda$ shrinks $\theta_{\min}$, so two close objects become distinct. This is why a TEM (very small de Broglie $\lambda$) out-resolves an optical microscope, and why a wider lens or telescope objective has higher resolving power.