Playground
Interactive single-slit diffraction: adjust slit width and wavelength to see the intensity pattern and central maximum width change.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Slit width Width of the single slit aperture | m | L | 0.00001 – 0.001 | |
| Angle to minimumoutput Angle from the central axis to the m-th dark fringe | rad | 1 | 0 – 0.5 | |
| Order number Integer order of the dark fringe (1, 2, 3, ...) | dimensionless | 1 | 1 – 10 | |
| Wavelength Wavelength of the incident light | m | L | 3.8e-7 – 7.5e-7 |
Deep dive
Derivation
Divide the slit of width a into infinitesimal strips. Each strip acts as a Huygens wavelet source. At angle θ, the path difference between the top and bottom of the slit is a·sin(θ). Dark fringes occur when this equals mλ (m = ±1, ±2, ...) because the slit can be divided into m pairs of strips that cancel pairwise. Formally, integrating the electric field contributions: E(θ) = E₀·sinc(πa·sinθ/λ), giving minima at a·sin(θ) = mλ.
Experimental verification
Grimaldi first observed (1665). Fraunhofer provided precise measurements using his diffraction grating. Modern: single-slit patterns are reproduced routinely with lasers and adjustable slits in undergraduate labs.
Common misconceptions
- The formula gives bright fringes — it gives dark fringes (minima). Bright maxima are approximately halfway between.
- Narrower slit means narrower pattern — the opposite: narrower slit produces wider diffraction spread.
- Diffraction and interference are different phenomena — diffraction IS interference from a continuous aperture.
Real-world applications
- Telescope resolution: the Airy disk is the 2D analog, setting the diffraction limit.
- CD/DVD reading: laser diffraction from pit tracks encodes data.
- X-ray crystallography: diffraction from atomic-scale 'slits' reveals molecular structure.
- Acoustic diffraction: sound bending around doorways and obstacles.
Worked examples
First dark fringe of a laser through a slit
Given:
- a:
- 0.00005
- lambda:
- 6.33e-7
- m:
- 1
Find: theta
Solution
θ = arcsin(mλ/a) = arcsin(1 × 6.33×10⁻⁷ / 5×10⁻⁵) = arcsin(0.01266) = 0.01266 rad ≈ 0.725°
Central maximum width on a screen
Given:
- a:
- 0.0001
- lambda:
- 5e-7
- L:
- 2
Find: width of central maximum
Solution
Width = 2λL/a = 2 × 5×10⁻⁷ × 2.0 / 1×10⁻⁴ = 0.02 m = 20 mm
Scenarios
What if…
- scenario:
- What if a >> λ?
- answer:
- Diffraction becomes negligible (θ → 0). Light passes in a straight beam — the geometric optics limit.
- scenario:
- What if a ≈ λ?
- answer:
- First minimum at θ ≈ 90°. Light spreads into a hemisphere — maximum diffraction.
- scenario:
- What if you use a circular aperture instead?
- answer:
- The pattern becomes an Airy disk: concentric bright and dark rings. The first dark ring is at sin(θ) = 1.22λ/D.
Limiting cases
- condition:
- a → ∞
- result:
- theta → 0
- explanation:
- Wide slit means negligible diffraction — geometric optics limit.
- condition:
- a → lambda
- result:
- theta → 90°
- explanation:
- Slit width equals wavelength: light spreads into a hemisphere.
- condition:
- lambda → 0
- result:
- theta → 0
- explanation:
- Short wavelength means sharp shadows — ray optics regime.
Context
Francesco Maria Grimaldi · 1665
Grimaldi first observed diffraction fringes. Fraunhofer later provided the mathematical treatment for far-field single-slit patterns.
Hook
Why does a laser beam spread out after passing through a narrow slit?
A 633 nm laser shines through a 0.05 mm slit. Find the angle to the first dark fringe using a*sin(theta) = m*lambda.
Dimensions:
- lhs:
- a·sin(θ) → [L]·[1] = [L]
- rhs:
- m·λ → [1]·[L] = [L]
- check:
- Both sides are [L] = meters. ✓
Validity: Valid in the Fraunhofer (far-field) regime where screen distance L >> a²/lambda. For near-field, use Fresnel diffraction theory.