PX275 - I7 - rayleigh criterion for resolution

to find the first dark fringe for an airy pattern
zero crossing of an airy pattern occurs at $x = 1.22 π / 2$

⟹ \frac{k ρ R}{D} = \frac{1.22 π}{2}

for small angles, $\sin θ \approx θ \approx ρ / D$

⟹ k R θ = \frac{1.22 π}{2}

the angular separation:

θ = \frac{1.22 π}{2 k R} = \frac{1.22 λ}{4 R}

this is the rayliegh criterion for diffraction limited resolution
the sources cannot be resolved if the central maxima overlap
the minimum angular separation that can be resolved is:

Δ θ > \frac{1.22 λ}{2 R}

where, the units are in radians

eg: eye, $R = 3$ mm, $λ = 500$ nm $⟹ Δ θ ≃ 20^{^{″}}$ (arcseconds)
eg: a telescope, $R = 0.5$ mm
- visible: $λ = 500$ nm $⟹ Δ θ ≃ {0.125}^{^{″}}$
- UV: $λ = 200$ nm $⟹ Δ θ ≃ {0.05}^{^{″}}$
- infrared: $λ = 10 μ$ m $⟹ Δ θ ≃ {2.5}^{^{″}}$