Optics

Diffraction of light

Diffraction is the spreading and redistribution of light at an aperture or obstacle; for a single slit, minima satisfy a sinθ = mλ.

a\sin\theta = m\lambda \quad (m=\pm1,\pm2,\ldots)

Definition: Single slit and the Huygens–Fresnel principle

Each point across the aperture acts as a secondary wave source; their superposition in the far field sets the intensity. For slit width a, the Fraunhofer field is E(θ) ∝ sinc(β), with β = πa sinθ/λ, and I/I₀ = (sinβ/β)². Minima occur at β = mπ for nonzero integer m.

View Huygens sources across the slit and the resulting Fraunhofer intensity; adjust λ, slit width a, and screen distance.

Model and quantities

Example: Estimating slit width from the central maximum

Light of wavelength λ = 600 nm passes through a slit of width a = 0.20 mm. A screen is L = 2.0 m away. Estimate the width of the central maximum between the first minima.

Solution

The first minima have sinθ ≈ θ = λ/a = 3.0×10⁻³ rad on either side. The half-width on the screen is x₁ ≈ Lλ/a = 6.0 mm, so the full central-maximum width is about 12 mm.

Quick check

Diffraction spreads light into the geometrical shadow when an aperture or obstacle is comparable in size to the wavelength. For a slit of width $a$ , minima approximately satisfy $a\sin heta=m\lambda$ ( $m=1,2,\ldots$ ); a narrower slit produces a wider central maximum. For example, a $0.10$ mm slit illuminated by $500$ nm light has its first minimum at about $0.29^\circ$ . Diffraction limits resolution: two nearby points become difficult to distinguish when their images overlap substantially.

For the same λ, a narrower single slit makes the central maximum on the screen:

In Fraunhofer single-slit diffraction, how does the first-minimum angle depend on slit width a?

References

Frank L. Pedrotti, Leno S. Pedrotti, Leno M. Pedrotti (2007). Introduction to Optics