Resolving Power (Rayleigh)

Rayleigh criterion: two point sources are just resolved when the central maximum of one falls on the first minimum of the other.

For a circular aperture (lens, telescope, eye): θ_min = 1.22 λ/D, where D = aperture diameter.

The 1.22 factor comes from the first zero of the Bessel function J₁ — different from a rectangular slit's λ/a.

Resolving power R = 1/θ_min. Larger D and shorter λ → better resolution.

Telescope resolving power: R_telescope = D/(1.22 λ). For Hubble (D = 2.4 m, λ = 550 nm): R ≈ 3.6 × 10⁶ rad⁻¹.

Microscope (Abbe limit): d_min = 1.22 λ/(2 NA), where NA = n sinα is the numerical aperture. Shorter λ and higher NA improve resolution.

Diffraction-limited optical microscopes can resolve ~200 nm with visible light. Electron microscopes use λ_e ~ 10⁻¹² m for atomic-scale resolution.