Diffraction and Diffraction Gratings
A single slit isn't infinitely narrow — treated as many point sources across its width, light from different parts of the slit can interfere with light from other parts of the same slit, producing a diffraction pattern with its own dark fringes. A diffraction grating pushes the double-slit idea the other direction: instead of 2 slits, use hundreds or thousands, which sharpens the bright fringes dramatically and is the working principle behind a spectrometer.
What you'll learn
- Calculate the angular position of dark fringes in single-slit diffraction.
- Explain single-slit diffraction as interference between different parts of the same slit.
- Calculate the angular positions of bright fringes (orders) produced by a diffraction grating.
- Relate a grating's line density to its slit spacing d.
- Determine the maximum diffraction order a grating can produce for a given wavelength.
- Explain why a diffraction grating produces much sharper bright fringes than a double slit.
1. Single-Slit Diffraction
A slit of finite width a, illuminated by a plane wave, doesn't produce a single sharp bright spot but a diffraction pattern — because different parts of the slit's own width act as separate sources that interfere with each other. This produces a bright central maximum flanked by progressively dimmer secondary maxima, with dark fringes at angles satisfying a sinθ=mλ.
Notice the structural resemblance to (and genuine difference from) the double-slit formula: here a is the width of a single slit, and this condition describes destructive, not constructive, interference.
2. Diffraction Gratings
A diffraction grating extends the double-slit idea to hundreds or thousands of evenly-spaced slits. The bright-fringe (order) condition, d sinθ=mλ, looks identical in form to the double-slit formula, but with many more sources contributing, each bright maximum becomes dramatically sharper: at the exact correct angle, all slits add perfectly in phase, but even a small deviation causes the many individual phase differences to accumulate and drive the sum toward near-total cancellation.
This sharpening (not just brightening) is exactly why gratings, not simple double slits, are the practical tool used in real spectrometers to measure wavelengths precisely.
Key equations
- a sinθ = mλ, m=1,2,3,... (dark fringes) — Single-slit diffraction dark-fringe condition — note this looks structurally like the double-slit *bright* fringe condition, but describes dark fringes here, and uses slit width rather than slit separation.
- d sinθ = mλ, m=0,1,2,... (bright fringes) — Diffraction grating bright-fringe (order) condition — structurally identical to the double-slit bright-fringe formula, but with d now referring to a grating's line spacing rather than two-slit separation.
- d = 1/N — Converts a grating's commonly-quoted line density into the spacing d used directly in the grating equation.