6.13 Double-Slit Interference and Diffraction Gratings

Double-slit interference

When coherent light passes through two narrow, closely spaced slits, overlapping waves create a pattern of bright and dark fringes on a screen. Bright fringes (constructive interference) occur where path lengths from the two slits differ by a whole number of wavelengths: d sinθ = mλ, where d is slit separation, θ is the angle to the fringe, and m = 0, ±1, ±2...

Dark fringes and fringe spacing

Destructive interference (dark fringes) occurs when the path difference is a half-integer number of wavelengths: d sinθ = (m + 1/2)λ. For small angles, fringe spacing on a screen a distance L away is approximately y = mλL/d. Smaller slit separation d or longer wavelength λ produces wider-spaced fringes.

Diffraction gratings

A diffraction grating has thousands of closely spaced slits, producing much sharper, brighter, and more widely separated maxima than a double slit, using the same equation d sinθ = mλ, where d is now the distance between adjacent grating lines. Gratings are used to precisely measure wavelengths and separate light into spectra, since different λ produce maxima at different angles.

Key terms

Double-slit bright fringe condition

d sinθ = mλ, where m = 0, ±1, ±2... (constructive interference)

Double-slit dark fringe condition

d sinθ = (m + 1/2)λ (destructive interference)

Fringe spacing formula

y = mλL/d, where L is the distance to the screen and d is slit separation

Diffraction grating

A device with many closely spaced slits that produces sharp, bright, well-separated interference maxima; same equation d sinθ = mλ applies

Effect of increasing wavelength on fringe spacing

Increases fringe spacing (fringes spread farther apart), since y ∝ λ

Try an AP-style question

Light of 500 nm passes through slits 0.10 mm apart onto a screen 2.0 m away. What is the fringe spacing?

1. 10 cm
2. 0.50 cm
3. 1.0 cm
4. 0.10 cm