§ 22. Single-Slit Wavefront Diffraction in 3D (Grade 11)
Wavefront spreading through a narrow slit, Huygens wavelets & sinc²(β) intensity profile. When a slit narrows toward the wavelength of light, the beam does not shrink, it fans out into a broad diffraction pattern.
Theory Summary & Scientific Mechanism
Diffraction is the bending and spreading of a wave as it passes through an aperture or around an obstacle, violating geometric ray optics whenever the aperture width a is comparable to the wavelength λ. By the Huygens–Fresnel principle, every infinitesimal strip across the open slit of width a radiates coherent secondary wavelets.
In the far-field (Fraunhofer diffraction) at angle θ, the path difference between wavelets from the two slit edges is Δ = asin θ. Dividing the slit into 2m equal zones shows that destructive cancellation occurs at diffraction minima asin θ = ± mλ (m = 1, 2, 3, …). Integrating across the continuous aperture gives the intensity profile I(θ) = I₀ ≤ft((sin β)/(β) )², where β = (π asin θ)/(λ).
Over 84% of the diffracted light power is concentrated in the central bright maximum bounded by sin θ₁ = ± (λ)/(a), with screen width W = (2λ L)/(a). This reveals a fundamental wave reciprocity: narrowing the slit a causes the diffracted beam on the screen to spread wider!
Everyday Analogy
If you shine a laser through a tiny slit between two razor blades, instead of a narrower dot you get a wider central streak with side fringes! Paradoxically, squeezing the slit narrower makes the diffraction pattern spread out wider!
Interactive Question & Derivation
A plane monochromatic light wavefront passes through a narrow vertical slit of width b. What happens to the width of the central bright maximum Δ y_central on the viewing screen if the slit width b is decreased?
- A: The central bright band on the screen narrows proportionally
- B: By the first diffraction minimum condition sin θ₁ = λ / b, narrowing the slit increases angular divergence and widens the central maximum (Δ y = 2λ L / b)
- C: The diffraction width depends only on the distance to the light source
Study Hint: Look at the first diffraction minimum condition sin θ₁ = λ / a: if slit width a decreases, does angle θ₁ grow wider or narrower?
Proof & Explanation: By the Huygens–Fresnel principle, secondary coherent wavelets across the slit interfere destructively at angles where bsin θ = ± mλ. The half-width of the central maximum is set by the first minimum (m=1): y₁ ≈ (λ L)/(b). Narrowing the slit b increases diffraction spreading and broadens the central peak Δ y = (2λ L)/(b)!
Key Equations & 3D Simulation Parameters
bsin θₘ = mλ, Δ y_central = (2λ L)/(b), I(θ) = I₀≤ft((sin β)/(β) )²
- Diffraction slit width b: 1.2 … 4.5 μm (default: 2.2 μm)
- Light wavelength λ: 400 … 700 nm (default: 550 nm)
- Screen distance L: 3 … 6.5 m (default: 4.5 m)
- Diffraction minimum order m: 1 … 3 (default: 1)