Section 9 — Diffraction
GE-4 / PHYS7021 unit 9 · 5 lecture-hours · 4 solved questions
9.1 Theory you need
Fresnel half-period zones: the wavefront is divided into zones of radii rn = √(nλb); adjacent zones differ by λ/2 in path. A zone plate blocks alternate zones — the remaining zones add in phase ⇒ it focuses light like a lens but with several focal lengths fn = r12/(nλ).
Grating equation: (a+b) sin θn = nλ , nmax = floor[(a+b)/λ]- Fresnel diffraction: source/screen near — curved wavefronts (straight edge, slit, wire patterns explained by half-period zones).
- Fraunhofer: source & screen effectively at infinity — single slit minima a sinθ = nλ; double/multiple slits give sharp principal maxima at (a+b) sinθ = nλ.
- Zone plate vs convex lens: both focus light; zone plate works by blocking (diffraction), has multiple foci, strong chromatic dispersion; lens works by refraction, single focus.
9.2 Solved past questions — exam-style answers
1. What is a zone plate? Explain its construction; compare with a convex lens.
- Construction: on a transparent plate draw concentric circles of radii rn = √(nλf) and blacken alternate annular zones; the remaining (transparent) zones send wavelets in phase to the focus f.
- It therefore behaves like a lens, forming a bright image of a distant source.
- Comparison with convex lens: (i) zone plate focuses by diffraction/blocking, lens by refraction; (ii) zone plate has several focal lengths f, f/3, f/5…, lens one; (iii) focal length of a zone plate depends strongly on λ (large chromatic aberration); (iv) image is less bright.
∴ Zone plate = diffraction "lens" with multiple foci f/n
2. Plane diffraction grating; grating constant/element; method of determining wavelength.
- A plane grating = a very large number of parallel, equidistant slits; grating element (a+b) = width of one slit + one opaque ruling; (a+b) = 1/N, N = lines per unit length.
- Principal maxima satisfy (a+b) sin θn = nλ.
- Method: mount the grating on a spectrometer table, illuminate normally with the light; measure the angle θn of a principal maximum in order n.
- Then λ = (a+b) sin θn/n.
∴ λ = (a+b) sinθₙ / n using spectrometer
3. Highest order spectrum: (2022) λ = 5.89×10−5 cm, 3000 lines/cm; (2023) λ = 5896 Å, 2000 lines/cm.
Principle: nmax = (a+b)/λ, since sinθ ≤ 1.
- 2022: (a+b) = 1/3000 = 3.33×10−4 cm; n = 3.33×10−4/5.89×10−5 = 5.66 ⇒ nmax = 5.
- 2023: (a+b) = 1/2000 = 5×10−4 cm; λ = 5896 Å = 5.896×10−5 cm; n = 5×10−4/5.896×10−5 = 8.48 ⇒ nmax = 8.
∴ 2022: 5th order ; 2023: 8th order
4. Grating numerical: λ = 500 nm, second-order maximum at 30° → lines per cm.
Given: n = 2, θ = 30°, λ = 500 nm = 5×10−5 cm. Formula: (a+b) = nλ/sinθ.
- (a+b) = (2 × 5×10−5)/sin 30° = (1×10−4)/0.5 = 2×10−4 cm.
- Lines per cm = 1/(a+b) = 1/(2×10−4) = 5000.
∴ 5000 lines per cm