Intensity in Single-Slit and Double-Slit Diffraction
Adding the contributions from every point across a slit gives the intensity as a function of angle, not only the positions of the minima. A real two-slit pattern is that single-slit envelope multiplied by the two-slit fringes.
What a learner can do afterwards
- Derives the single-slit intensity pattern by adding contributions across the slit
- Explains missing orders in a two-slit pattern using the diffraction envelope
- Predicts how the pattern changes when the slit is widened
1 · Read
Picture the slit as a row of tiny sources side by side. Each sends a wavelet, and at any screen angle their crests arrive with shifted phases. Add them as phasors head to tail: a straight line at center, a curling spiral off center. The curl is summed up in beta, pi times slit width times sine of angle over wavelength, and the brightness is the central brightness times sine of beta over beta, squared.
The spiral closes into full circles wherever beta is a whole multiple of pi, which is exactly where slit width times sine of angle equals a whole number of wavelengths. Those dark angles fence in a bright central peak about two wavelengths over slit width wide. Widen the slit and the peak narrows: with 550 nm light and a 2.00 micrometer slit the first dark angles sit at 16.0 and 33.4 degrees.
With two slits the interference fringes, separation times sine equals order times wavelength, get dimmed by the single-slit envelope. Where a bright fringe lands on an envelope dark spot, that order goes missing: with width twice and separation six times the wavelength, orders plus and minus 3 vanish, and the same repeats at multiples. With 0.020 mm slits 0.20 mm apart in 500 nm light, orders plus and minus 10 vanish, and again at 20, leaving 19 bright fringes in the central peak.
Nearly all the light sits in the central peak: off center the phasors curl almost one and a half turns, so the leftover sum is short and the first side bump reaches only about 0.045 of the center. When you work both equations at once, give the two whole numbers different letters so orders and minima never mix.
Wavelets across each slit sum to an envelope that dims the two-slit fringes and deletes whole orders.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.