--- title: "Intensity in Single-Slit and Double-Slit Diffraction" description: "Continuous phasor addition gives the far-field intensity of a uniformly illuminated slit. For two identical slits under coherent monochromatic normal illumination with common polarisation, the single-" canonical: https://lightmysky.com/learn/science/intensity-in-single-slit-and-double-slit-diffraction-mt_HURhzZdIkM source: https://lightmysky.com/learn/science/intensity-in-single-slit-and-double-slit-diffraction-mt_HURhzZdIkM.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Intensity in Single-Slit and Double-Slit Diffraction Continuous phasor addition gives the far-field intensity of a uniformly illuminated slit. For two identical slits under coherent monochromatic normal illumination with common polarisation, the single-slit envelope multiplies the interference factor. Their overlap can suppress an otherwise allowed interference order. Subject: Science · Area: Waves, Light & Sound · Ages 20 to 21 Page: https://lightmysky.com/learn/science/intensity-in-single-slit-and-double-slit-diffraction-mt_HURhzZdIkM ## Ready when they can - 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 ## Lesson: What sets the brightness across a slit pattern Add phasors head to tail. Radian angle = arc length/radius. For a uniformly illuminated slit with monochromatic normal illumination in the far field, beta = pi a sin(theta)/lambda; a is width and theta is from the beam direction. Signed phase spread is 2beta radians. For beta ≠ 0, the arc has radius r > 0 in field-amplitude units, traced length E0 = 2r|beta| and chord magnitude E = 2r|sin(beta)|. Thus E/E0 = |sin(beta)/beta| and I/I0 = [sin(beta)/beta]^2. The arc turns through 2beta radians. Half-angle circle geometry gives chord magnitude 2r|sin(beta)|. E0 and I0 are the single slit's central field-amplitude magnitude and intensity. At beta = 0, direct substitution gives 0/0, not zero. The ratio |sin(beta)/beta| approaches 1 as beta approaches zero, so central intensity is I0. At the centre, aligned phasors add in a straight line without an arc radius. *(drawing: These equations use nonnegative lengths. r is the phasor-circle radius in field-amplitude units. Beta is in radians. The cells do not draw the arc.)* A single-slit minimum needs sin(beta) = 0 with beta nonzero. Thus beta = p pi and a sin(theta) = p lambda for p = ±1, ±2, and so on. Zero is excluded because it is the central maximum. A stated minimum also needs |p lambda/a| ≤ 1 so that its angle exists. When a exceeds lambda, the central boundaries are theta = ±arcsin(lambda/a). At small angles their full separation is approximately 2lambda/a radians. Increasing a at fixed wavelength narrows this peak. For 550 nm light and a = 2.00 micrometres, the first minima are about ±16.0 degrees. The second minima are about ±33.4 degrees, outside the central peak. For identical slits of width a and centre spacing d, use uniform coherent monochromatic normal illumination, common polarisation and far-field observation. Define I2,0 as the combined central intensity: I2,0 = 4I0, where I0 is one slit's central intensity. Then I/I2,0 = [sin(beta)/beta]^2 cos²(alpha), with alpha = pi d sin(theta)/lambda. Missing orders require integer m = p d/a for nonzero integer p, with |m lambda/d| ≤ 1. *(drawing: Use the stated two-slit model with a = 4lambda and d = 20lambda. The first missing orders have sin(theta) = ±1/4, so their angles exist.)* For the same two-slit model with a > lambda, the first envelope minima have sin(theta) = ±lambda/a. Integer interference orders strictly inside this central envelope satisfy |m| < d/a. Count negative orders, zero once, and positive orders. If d/a is an integer R, there are 2R − 1 such orders; boundary orders ±R are missing. Check physical angles as well. At beta = 3pi/2 radians, the phasor arc makes one and a half turns. The chord is a diameter, not a nearly closed endpoint gap. Its ratio to arc length is 2/(3pi). Squaring gives I/I0 = 4/(9pi²), about 0.045. This is a value near the first side-lobe maximum, not its exact maximum. For each nonzero integer envelope order p, calculate the candidate m = p d/a. It is a missing interference order only if m is an integer and |m lambda/d| ≤ 1. A noninteger candidate is not an interference order. An integer candidate with required sine above 1 has no physical angle, rather than an envelope-suppressed direction. Apply both checks in the stated identical-slit far-field model. **Recap.** Divide the chord magnitude by the traced phasor length and square. Nonzero envelope minima can suppress integer, accessible interference orders in the stated two-slit model. *Derivation, diffraction-envelope teaching and the 550 nm example adapted from OpenStax's University Physics Volume 3. Revised questions by the dataset authors.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Phasor Addition and the Mathematics of Interference](https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw) - [Single-Slit Diffraction and the Limit of Resolution](https://lightmysky.com/learn/science/single-slit-diffraction-and-the-limit-of-resolution-mt_qiG0fGadBU) ## Optional connections - [The Diffraction Grating and Measuring Wavelength](https://lightmysky.com/learn/science/the-diffraction-grating-and-measuring-wavelength-mt_yIjB-uJuFt) ## Opens up - [X-Ray Diffraction and the Bragg Condition](https://lightmysky.com/learn/science/x-ray-diffraction-and-the-bragg-condition-mt_qEDplGeIkB)