Single and double slit

Single and double slit

1. Single Slit (Fraunhofer) — Width $a$

Plane wave hits slit $-a/2 ≤ x ≤ a/2$, far-field amplitude:

,

Intensity:

• Minima (dark):

(not $n=0$ — central max there)

• Maxima: approximate at

⇒ $α≈1.43π,2.46π$; central max ($θ=0$) intensity $I_0$, side maxima ~4.5%,1.6% falling.

• Width: central max spans $-λ/a < sinθ < λ/a$ (first minima). If white light, colours separate.

2. Double Slit — Two Slits Width $a$, Separation $d$ (centre-centre)

Amplitude = single-slit envelope × two-slit interference:

,

Intensity:

• Interference maxima (bright fringes):

(cos=±1)

• Diffraction minima (envelope zeros, missing orders):

⇒ if ratio $d/a$ integer, some interference maxima coincide with diffraction minima and are missing (suppressed).

Single slit: broad envelope, first minima at a sinθ=λ, central intensity I0
Double slit: sharp cos² fringes inside same envelope, fringe spacing Δ(sinθ)=λ/d (finer if d>a), envelope controls brightness
If a≪d, many interference fringes inside central envelope; if a≈d, few.

5. Exam Points

• Derive $I=I_0[sinα/α]²$ and minima $a sinθ=nλ$ for single slit — 7 marks, very frequent.
• Derive double slit $I=4I_0 cos²β[sinα/α]²$, maxima $d sinθ=mλ$ and missing orders when $d/a$ integer — 7-10 marks, very frequent.
• Sketch intensity vs $θ$ showing envelope — 5 marks, frequent.