Diffraction grating · resolving power

Add more slits to Young's two — why do the soft fringes sharpen into spectral lines, and what sets whether a grating can split two almost-identical colours?

▶ Run the simulationSee the measured result

Units: metres (fed doublet splitting 0.60 nm; NIST Na D₂ 588.99509, D₁ 589.59244, Δλ = 0.59735 nm — the module rounds each line to 0.1 nm)

How the lab tests it

Model the N-slit grating factor [sin(Nγ)/(N sinγ)]² (γ=πd·sinθ/λ) for the sodium doublet (D₂ 589.0 nm, D₁ 589.6 nm) through a 600-line/mm grating with N=600 lines illuminated; scan a noisy intensity profile across the 1st- and 2nd-order peaks, smooth and count the local maxima, and from the two 2nd-order peak angles recover each wavelength via λ=d·sinθ/m.

What it checks

the grating equation d·sinθ=mλ fixes the orders (here 0,±1,±2 — the 3rd needs sinθ>1 and is evanescent), and N slits squeeze each into a line of width ∝1/N. That sharpness sets the RESOLVING POWER R=λ/Δλ=m·N: the same 600-line grating fails to split the sodium doublet (Δλ=0.6 nm, R_needed≈982) in 1st order (R=600, one merged line) but RESOLVES it in 2nd order (R=1200, two lines) — resolution grows with BOTH line count and order. The lab recovers λ≈589.0 & 589.6 nm and Δλ≈0.6 nm from the noisy 2nd-order angles. The N-slit sequel to Young's two-slit comb: from weighing a wavelength to telling two apart.

Diffraction grating, resolving power & grating-constant calculator

Two slits weigh a colour; six hundred of them tell two colours apart. That is the whole step this page is about, and it is not a step in brightness. Adding slits leaves the orders exactly where Young's pair put them — d·sin θ = mλ, an ARCSINE with no screen distance in it anywhere — and does something else instead: it makes each line thinner, as λ/(Nd), which falls as 1/N. Thin lines are what a spectrometer is for, and the number that decides whether two colours arrive as one line or two is Rayleigh's resolving power R = λ/Δλ = m·N, in which the slit COUNT and the ORDER appear and the intensity does not. The simulation above is the demonstration, run on the sodium doublet: the same 600-line grating, the same lamp, one merged line in first order where R = 600 and two clean lines in second where R = 1200, because 982 is what that pair demands. The spacing is never typed here — it is inverted out of a ruling density, or recovered from a line you already know, which is the order of operations in a real laboratory: calibrate against a sodium lamp, then point the instrument at a spectrum you do not know. One box carries the rival. p is the superposition exponent, 2 when amplitudes add and are then squared and 1 when intensities add — Newton's corpuscles — and N^(p−1) is the peak-to-mean: 600 against 1, which is to say a spectrum against a blank screen. The measured-angle box holds this lab's own reading rather than a tidy one, so inverting it returns 588.9833333 nm and the page prints the −28.30 ppm. And the last direction REPLAYS the simulation's own measurement — the same scan, the same self-seeded noise, the same smoothing and peak test — lands on its executed values bit for bit, and then decomposes the +4.17% by which the number on screen exceeds the splitting fed in, by switching one stage of that code off at a time: the two peaks are pushed apart by each other's first sidelobe, and neither the smoothing nor the noise moves them at all. Four things this page will not do: model a blaze angle or finite slit width (the simulation's slits are points, so there is no envelope and no ghost orders), overlap orders of a white-light spectrum, reproduce the D₂:D₁ ≈ 2:1 intensity ratio of real sodium (the lines are fed equal), or correct the simulation above — the +4.17% is disclosed on its own HUD, beside the number it belongs to.