Resolving Power (Rayleigh)
θ_min = 1.22 λ/D. Two Airy patterns transition resolved → unresolved.
Key Notes
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.
Formulas
Rayleigh criterion (circular)
Minimum resolvable angle for a circular aperture.
Rayleigh criterion (rectangular slit)
Slit aperture — no 1.22 factor (different Bessel/sinc geometry).
Microscope resolution (Abbe)
Minimum resolvable separation in object plane.
Telescope resolving power
Reciprocal of θ_min — larger D and shorter λ both help.
Eye resolving angle
Limited by pupil diameter (~3 mm) and retinal cone spacing.
Important Points
What matters for resolution is the APERTURE, not the magnification. A bigger lens resolves more — magnifying a blurry image only blurs more.
UV microscopes and X-ray microscopes give better resolution thanks to shorter λ — but materials become opaque at short λ.
Electron microscopes exploit de Broglie λ ≈ h/√(2mE). At 100 keV, λ_e ≈ 0.004 nm — atomic resolution possible.
Super-resolution techniques (STED, PALM, STORM) beat the Rayleigh limit by switching individual fluorophores — Nobel 2014.
JEE often asks for minimum distance between two objects resolvable by a telescope/microscope given λ and D — apply Rayleigh directly.
Resolving Power (Rayleigh) notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.