A finite optical aperture cannot form a perfect geometrical point image of a point source. Because light diffracts at the aperture, a distant point source produces a central bright spot surrounded by weaker rings rather than an infinitesimal point.

For a circular aperture of diameter $D$, the first dark ring of this Airy pattern occurs at an angle approximately

$$\boxed{\theta_{\min}\approx1.22\frac{\lambda}{D}}$$

when the angle is small and measured in radians.

This angular scale sets a fundamental diffraction limit: increasing aperture diameter makes the pattern narrower, while increasing wavelength makes it broader.

Resolving two point sources

Two nearby sources produce overlapping diffraction patterns. The Rayleigh criterion provides a conventional boundary for when they are just resolved: the central maximum of one Airy pattern lies approximately at the first minimum of the other.

For a circular aperture, their minimum angular separation is therefore

$$\boxed{\theta_R\approx1.22\frac{\lambda}{D}}.$$

Sources separated by much more than this angle are readily distinguishable in an ideal diffraction-limited system. Sources separated by much less produce strongly merged patterns.

The Rayleigh criterion is a useful convention, not a statement that information changes discontinuously at exactly one angle.

Worked example: telescope aperture

A telescope has diameter

$$D=0.20,\mathrm m$$

and observes light of wavelength

$$\lambda=500,\mathrm{nm}=5.00\times10^{-7},\mathrm m.$$

Its Rayleigh angular resolution is

$$\theta_R \approx1.22\frac{5.00\times10^{-7}}{0.20} =3.05\times10^{-6},\mathrm{rad}.$$

Using

$$1,\mathrm{rad}\approx206265\ \text{arcseconds},$$

this is

$$\theta_R\approx0.63\ \text{arcseconds}.$$

So under ideal diffraction-limited conditions, two equally bright point sources separated by about $0.63$ arcseconds are near the conventional resolution limit at this wavelength.

How aperture and wavelength affect resolution

Because

$$\theta_R\propto\frac{\lambda}{D},$$

a larger aperture gives smaller angular separation and therefore better resolution.

For example, doubling $D$ halves the diffraction-limited angle. Likewise, observing at half the wavelength halves the angle if the same aperture can be used effectively.

This is why large telescope mirrors and lenses improve angular resolution in addition to collecting more light.

From angular to spatial resolution

If two objects are a distance $L$ away and subtend a small angular separation $\theta$, their transverse separation is approximately

$$s\approx L\theta.$$

Thus an angular diffraction limit can be translated into a minimum distinguishable spacing at a known distance.

Real optical systems

Diffraction is only one limitation. Atmospheric turbulence, lens aberrations, detector sampling, motion and noise can make the practical resolution worse than the ideal value. Techniques such as adaptive optics can reduce some of these additional limitations but cannot remove diffraction from a finite aperture.

The key distinction is between magnification and resolution. Making an already blurred diffraction pattern larger does not reveal arbitrarily fine detail. True diffraction-limited resolution improves by increasing the effective aperture or using a shorter wavelength.