& Circular aperture

& Circular aperture

1. What Changes vs Rectangular Slit

A circular aperture radius $a$ (diameter $D=2a$) gives a 2D diffraction pattern with circular symmetry: central bright Airy disc surrounded by concentric dark/bright rings (Airy pattern), not straight fringes.

Huygens: integrate over circular area → Bessel function $J_1$, not sinc.

2. Intensity — Airy Pattern

For Fraunhofer, plane wave, lens focal plane:

where

, $θ$ is angle from axis, $J_1$ is Bessel first order.

• First dark ring (first minima):

(the famous $1.22$ factor). Derivation: $q=3.832 ⇒ D sinθ≈1.22λ$.

• Central Airy disc contains ~84% of total power; radius to first dark:

on screen at focal length $f$ (or $1.22λ/distance$ angular).

• Subsequent minima at $q≈7.016,10.174$ ⇒ faint rings 1.7%,0.4% intensity.

3. Comparison to Single Slit

Slit (1D)	Circular (2D)
$I∝[sinα/α]²$, minima $a sinθ=nλ$	$I∝[2J_1(q)/q]²$, first minima $D sinθ=1.22λ$
central width $2λ/a$ (angular)	Airy radius $1.22λ/D$
4.5% first side max	1.7% first ring

4. Why It Matters — Resolution Limit

A lens cannot focus to a point — even a perfect lens gives an Airy disc size $≈1.22λ/D$ (or $1.22λ/NA$). This is the diffraction limit leading to Rayleigh criterion next topic.

5. Worked Examples

(a) Telescope aperture D=0.1 m, λ=550 nm: θ_1=1.22·5.5e-7/0.1≈6.7 µrad (≈1.4 arcsec) — smallest resolvable separation ✓
(b) Camera f/2.8: effective D? Airy disc diameter ≈1.22λ·f/#? For photography, Airy disc ~1.35·λ·N? Actually 1.22λ·f/D ⇒ for N=f/D.
(c) Microscope: objective NA≈1.4, λ=500 nm: spot ≈0.61λ/NA≈0.22 µm (Abbe) ✓

(d) Why pinhole camera not sharp? Small D ⇒ large Airy disc dominates geometric spot.

6. Exam Points

• Derive/introduce Airy pattern $I=I_0[2J_1(q)/q]²$ and $sinθ_1=1.22λ/D$ — 5-7 marks, very frequent.
• Sketch Airy disc with rings, state 84% energy in central — 3-5 marks, frequent.
• Compare to single slit — 3 marks.
• Link to resolution next topic — 3 marks.