Rayleigh criterion

Rayleigh criterion

1. What It Is

Rayleigh criterion = practical rule for resolvability of two close point objects whose diffraction patterns (Airy discs) overlap. Two images are just resolved when central maximum of one Airy disc coincides with first minima of the other.

For circular aperture diameter $D$, wavelength $λ$:

is the minimum angular separation to be just resolved.

For microscope, using linear separation $y_{min}$ at object distance with numerical aperture $NA = n sinα$:

is the Abbe/Rayleigh limit (often $0.61$ comes from $1.22/2$).

2. Why 1.22?

From circular aperture first dark $sinθ≈1.22λ/D$. Rayleigh says resolution needs at least that separation — arbitrary but convenient (dip 26% between peaks, eye can see).

Two Airy discs: separation >>θ_R ⇒ two peaks distinct (resolved)
               ≈θ_R ⇒ central max of one at first dark of other ⇒ dip ~26% (just resolved)
               <<θ_R ⇒ merged single peak (unresolved)

5. Exam Points

• State Rayleigh criterion $θ=1.22λ/D$ or $0.61λ/NA$ — 3-5 marks, very frequent.
• Sketch two Airy discs at just resolved — 3 marks, frequent.
• Compute telescope/microscope/eye resolution — 5 marks, very frequent.
• Mention 26% dip — 3 marks.