How do I find the wavelength from the fringe spacing?

Rearrange w = λD/d to λ = wd/D. Fringes 4.4 mm apart from slits 0.25 mm apart on a screen 2.0 m away give λ = 4.4 × 10⁻³ × 0.25 × 10⁻³ / 2.0 = 5.5 × 10⁻⁷ m, which is 550 nm (green). Pick Wavelength λ under Solve for to see the working.

What is the formula for fringe spacing?

For small angles the bright fringes are equally spaced by w = λD/d, where λ is the wavelength, D the slit to screen distance and d the slit separation. With λ = 550 nm, D = 2 m and d = 0.25 mm, w = 4.4 mm.

Why are diffraction grating maxima so sharp?

With N slits, light from all N must arrive in step to give a maximum, and a tiny change in angle already puts the slits out of step with each other. The width of each maximum shrinks roughly as 1/N while the peak intensity grows as N², so a grating with thousands of lines gives very sharp, bright lines.

How many orders can a grating produce?

sin θ cannot exceed 1, so the highest order is nₘₐₓ = d/λ rounded down. A 600 lines per mm grating has d = 1.667 μm; with 500 nm light that gives 3.33, so orders up to n = 3 appear on each side.

Why do some orders go missing?

Each slit has a width a, and its single-slit pattern is zero where a sin θ = λ, 2λ and so on. If d/a is a whole number, say 3, then the 3rd, 6th and later orders land exactly on those zeros and disappear.

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Double slit and diffraction grating

Light through two narrow slits spreads out and overlaps with itself. Where the waves arrive in step you see a bright fringe; where they cancel, a dark one.

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What's happening

Each slit acts as a new source of waves, and the two sets of waves are coherent because they come from the same light. At a point on the screen the waves have travelled different distances. If that path difference is a whole number of wavelengths, d sin θ = nλ, crests meet crests and the light is bright; halfway between, crests meet troughs and it is dark. For small angles the bright fringes are equally spaced, w = λD/d, so longer wavelengths and a more distant screen spread them out, while wider-spaced slits squeeze them together. Add more slits and the bright lines stay in the same places but become much narrower and brighter: that is a diffraction grating, used to measure wavelengths precisely. Each slit also has a width a, and its own single-slit pattern acts as an envelope over the fringes; when d/a is a whole number some orders fall exactly on a dark point of the envelope and go missing.

d sin θ = nλw = λD / dnₘₐₓ = ⌊d / λ⌋I = I₀ sinc²(πa sin θ / λ) [sin(Nφ/2) / N sin(φ/2)]²

A-Level Physics (AQA, Edexcel, OCR): Young's double-slit experiment, w = λD/s or λD/d, path difference and the diffraction grating d sin θ = nλ. First-year university optics: N-slit interference and the single-slit envelope.

Work through the numbers with Physics Formulas and Scientific Calculator.

Challenge

Predict first: green light (λ = 550 nm) passes through two slits 0.25 mm apart onto a screen 2.0 m away. How far apart are the bright fringes? Then switch to a grating with 600 lines per mm: at what angle is the first-order maximum?

FAQ

How do I find the wavelength from the fringe spacing?: Rearrange w = λD/d to λ = wd/D. Fringes 4.4 mm apart from slits 0.25 mm apart on a screen 2.0 m away give λ = 4.4 × 10⁻³ × 0.25 × 10⁻³ / 2.0 = 5.5 × 10⁻⁷ m, which is 550 nm (green). Pick Wavelength λ under Solve for to see the working.
What is the formula for fringe spacing?: For small angles the bright fringes are equally spaced by w = λD/d, where λ is the wavelength, D the slit to screen distance and d the slit separation. With λ = 550 nm, D = 2 m and d = 0.25 mm, w = 4.4 mm.
Why are diffraction grating maxima so sharp?: With N slits, light from all N must arrive in step to give a maximum, and a tiny change in angle already puts the slits out of step with each other. The width of each maximum shrinks roughly as 1/N while the peak intensity grows as N², so a grating with thousands of lines gives very sharp, bright lines.
How many orders can a grating produce?: sin θ cannot exceed 1, so the highest order is nₘₐₓ = d/λ rounded down. A 600 lines per mm grating has d = 1.667 μm; with 500 nm light that gives 3.33, so orders up to n = 3 appear on each side.
Why do some orders go missing?: Each slit has a width a, and its single-slit pattern is zero where a sin θ = λ, 2λ and so on. If d/a is a whole number, say 3, then the 3rd, 6th and later orders land exactly on those zeros and disappear.