No optical instrument can resolve arbitrarily fine detail, even if its lenses or mirrors are perfectly shaped. The reason is
diffraction. When light passes through a finite circular aperture, it does not form a perfect point image. Instead, it spreads into a diffraction pattern with a bright central spot surrounded by weaker rings. For a circular aperture, this pattern is called the Airy pattern.
Because each point source produces an Airy pattern rather than an exact point, two nearby sources can overlap strongly and become difficult to distinguish. A standard criterion for deciding when two circular-aperture images are “just resolved” is the
Rayleigh criterion:
\[
\theta_{\min}=\frac{1.22\lambda}{D},
\]
where \(\lambda\) is the wavelength of light and \(D\) is the aperture diameter. The result \(\theta_{\min}\) is the minimum angular separation, in radians, that can be resolved according to the Rayleigh rule.
The factor \(1.22\) comes from the first minimum of the Airy diffraction pattern for a circular aperture. In the Rayleigh limit, the central maximum of one point source falls at the first minimum of the other. This does not mean the two images are perfectly separated, but it gives a very widely used practical threshold.
If you know the distance \(L\) to the target, then the angular resolution can be converted into an approximate
linear resolution using the small-angle relation
\[
s_{\min}\approx \theta_{\min}L.
\]
This tells you the smallest separation on the target itself that the aperture can distinguish at that distance.
The approximation is valid because \(\theta_{\min}\) is usually a very small angle.
For the sample case of a telescope with aperture \(D=0.2\ \text{m}\) and wavelength \(\lambda=550\ \text{nm}\), the wavelength must first be converted into meters:
\[
\lambda=550\times 10^{-9}\ \text{m}=5.50\times 10^{-7}\ \text{m}.
\]
Then the Rayleigh criterion gives
\[
\theta_{\min}=\frac{1.22(5.50\times10^{-7})}{0.2}
\approx 3.36\times10^{-6}\ \text{rad}.
\]
This is a very small angle, which is why telescope resolution is often reported in arcseconds rather than degrees.
Since \(1\ \text{rad}\approx 206265\ \text{arcsec}\), the sample value corresponds to about
\[
\theta_{\min}\approx 0.69\ \text{arcsec}.
\]
A larger aperture improves resolution because \(D\) appears in the denominator. If you double the aperture, you halve the Rayleigh limit and can distinguish smaller angles. A shorter wavelength also improves resolution for the same reason.
That is why blue light gives a slightly smaller diffraction limit than red light, all else equal.
The Airy pattern viewpoint is important conceptually. Even if the geometric-optics rays from two stars would appear distinct, diffraction can make their central spots overlap enough that they merge visually. The calculator’s preview separation factor
\(\theta_{sep}/\theta_{\min}\) is meant to show exactly that transition: below the Rayleigh limit the overlap is strong, near the limit the patterns are barely distinguishable, and above the limit the separation becomes clearer.
At more advanced levels, other resolution criteria exist, such as the Sparrow limit or full modulation-transfer analysis. Real systems are also affected by atmospheric turbulence, optical aberrations, sensor sampling, and signal-to-noise limits.
But the Rayleigh formula remains one of the most important first estimates in optics because it captures the fundamental diffraction barrier of a circular aperture:
\[
\theta_{\min}\propto \frac{\lambda}{D}.
\]
So the big lesson is simple: better resolution comes from shorter wavelength and larger aperture, and the reason is not imperfect glass alone but the wave nature of light itself.
