Diffraction Grating

1. What a Grating Is

Diffraction grating = array of many equally spaced parallel slits (typically $N$ = 5000–60000 lines total, ruling spacing $d$ = grating element $= a+b$ where $a$ = slit width, $b$ = opaque). It works as multiple-beam interference (many slits) enveloped by single-slit diffraction.

Types: transmission (glass ruled) and reflection (metal ruled); plane vs concave.

2. Grating Equation (Fraunhofer, normal incidence)

For incidence $θ_i$ (from normal) and diffraction angle $θ_m$ for order $m$ :

d (sinθ_m - sinθ_i)= m λ

For normal incidence ( $θ_i=0$ ):

d sinθ_m = m λ , m=0,±1,±2,...

$m$ = order (0th central, ±1st, etc.).
For oblique incidence at $θ_i$ , replace with $θ_i$ offset.
Max order limited by $|sinθ_m|<1$ ⇒ $|m|<d/λ$ .

Intensity: sharp principal maxima at those angles, intensity $∝ N²$ at peak, width $∝1/N$ (narrower with more lines).

3. Why Grating Gives Spectroscopy

Different $λ$ diffract to different $θ_m$ for same $m$ ⇒ disperses white light into spectrum much more than prism, with $θ$ increasing with $λ$ . Angular dispersion $dθ/dλ = m/(d cosθ_m)$ . Higher order gives larger dispersion but overlapping orders (e.g., $m=1, λ=600$ nm and $m=2, λ=300$ nm same $θ$ ).

4. Worked Examples

(a) Grating with 5000 lines/cm ⇒ d=2 µm: for λ=600 nm, first order sinθ=0.3 ⇒ θ≈17.5°, second sinθ=0.6 ⇒ 36.9°, third sinθ=0.9 ⇒ 64°; fourth would need sinθ=1.2 impossible ⇒ max order 3 ✓

(b) Need to resolve λ=589.0 and 589.6 nm (Na doublet) with m=1, d=2 µm: require resolving power R=mN = λ/Δλ=589/0.6≈982 ⇒ N≈982 lines needed ⇒ easy with grating 10 mm wide (5000 lines/cm gives 5000) ✓

(c) Missing orders: if a = b (equal slit/opaque), then d=2a ⇒ condition a sinθ=nλ gives missing grating orders when m =2n (even orders suppressed) — similar to double slit missing.

(d) Grating vs prism: grating dispersion ∝1/d, prism ∝dn/dλ material; grating more uniform.

5. Exam Points

State and derive grating equation $d sinθ=mλ$ — 5 marks, almost certain.
Find $θ_m$ , max order, grating element $d=1/(\text{lines per unit})$ — 5-7 marks, very frequent.
Dispersion and grating as spectrometer — 5 marks, frequent.
Missing orders — 3-5 marks, occasional.

Diffraction Grating