Diffraction Gratings, Dispersion & Resolving Power

Send a plane wave at normal incidence onto N slits of spacing d and collect the light at angle θ, far enough away that the rays are parallel. Slit j is d sin θ further from the screen than slit j − 1 for every j, so its field lags by the same δ = 2π d sin θ/λ. The total is E₀(1 + e^(iδ) + e^(2iδ) + … + e^(i(N−1)δ)), a geometric series with sum E₀ (e^(iNδ) − 1)/(e^(iδ) − 1); factoring a half-angle out of each bracket leaves the magnitude E₀ sin(Nδ/2)/sin(δ/2). Square it: I = I₀ sin²(Nδ/2)/sin²(δ/2). At δ = 2πm both sines vanish and the ratio tends to N, so the principal maxima reach N² I₀ — N phasors laid end to end in a straight line. Between them the numerator vanishes N − 1 times, at δ = 2πm + 2πp/N with p = 1 … N − 1, where the phasors close into a regular polygon; between those zeros sit N − 2 secondary maxima. For N = 5 the first secondary maximum, near δ = 104°, is 1.56 I₀ against a principal 25 I₀, some 6%; for large N it settles at (2/(3π))² = 4.5%. Check N = 2: sin²δ/sin²(δ/2) = 4 cos²(δ/2), the double-slit result.

The position of a principal maximum, d sin θ = mλ, has no N in it: a grating and a double slit of the same spacing throw their orders to the same angles. What N buys is width. The first zero beside the mth maximum sits at δ = 2πm + 2π/N, a phase step of 2π/N. Since dδ = (2π d cos θ/λ) dθ, that is an angle Δθ = λ/(N d cos θ) = λ/(W cos θ), with W = Nd the width of grating actually lit. Take d = 2.00 μm and λ = 500 nm: the first order lies at θ = 14.5°, and with N = 2 the half-width is 0.129 rad, or 7.4°, a fringe rather than a line. Light 20 mm of the same pitch, N = 10 000, and it shrinks to 2.6 × 10⁻⁵ rad, about 5 arcseconds, while the peak intensity has grown by (10 000/2)² = 2.5 × 10⁷. Height × width goes as N² × 1/N = N, in step with the light N slits admit: the sharpening costs no energy, it only pulls light out of the gaps between orders.

Hold m fixed and differentiate d sin θ = mλ: d cos θ dθ = m dλ, so the angular dispersion is D = dθ/dλ = m/(d cos θ). Substitute m/d = sin θ/λ and it becomes tan θ/λ — the dispersion depends only on where the line lands, not on the pitch or order that put it there. A 600 lines mm⁻¹ grating, d = 1.667 μm, sends 589 nm to 20.7° in first order, where D = 1/(1.667 μm × cos 20.7°) = 0.641 mrad nm⁻¹; in second order the line sits at 45.0° and D = 1.70 mrad nm⁻¹. The sodium D pair, 0.597 nm apart, is therefore separated by 0.38 mrad and 1.01 mrad, which a 300 mm focusing lens turns into 0.115 mm and 0.304 mm at the detector. Third order does not exist here: m cannot exceed d/λ = 2.83. Tilt the incident beam by θᵢ and every order shifts according to d(sin θᵢ + sin θₘ) = mλ, which is how reflection gratings reach orders that normal incidence cannot; the dispersion formula is unchanged.

Two lines at λ and λ + Δλ are resolved, by Rayleigh's convention, when the peak of one falls on the first zero of the other. Dispersion sets their angular separation, (m/(d cos θ)) Δλ; the half-width of a line is λ/(N d cos θ). Equate them and cos θ cancels: Δλₘᵢₙ = λ/(mN). The chromatic resolving power is R = λ/Δλₘᵢₙ = mN, order times illuminated slit count, and the summed profile of two lines at that limit dips to 8/π² = 81% of the peaks. Rewrite it with the grating equation: mN = N d sin θ/λ = W sin θ/λ. Resolving power is the illuminated width measured in wavelengths, scaled by sin θ ≤ 1, so W/λ is a ceiling no choice of pitch or order can beat; oblique incidence, where mλ = d(sin θᵢ + sin θₘ), only stretches it to 2W/λ. The 600 lines mm⁻¹ grating lit across 25 mm has N = 15 000 and R = 15 000 in first order, Δλₘᵢₙ = 0.039 nm at 589 nm: the D pair is fifteen times over resolved, but two lines 0.02 nm apart need second order, R = 30 000, or a wider beam.

Each order is a full spectrum, and they fan out together. Order m of wavelength λ + Δλ lands where order m + 1 of λ does when m(λ + Δλ) = (m + 1)λ, so the clean span in order m, the free spectral range, is Δλ_FSR = λ/m. In first order at 400 nm that is 400 nm: nothing visible overlaps, because second-order 400 nm arrives with first-order 800 nm, in the infrared. In second order the span at 400 nm is 200 nm, and third-order 400 nm coincides with second-order 600 nm, violet on orange, so any spectrometer working above first order carries an order-sorting filter or a cross-dispersing prism. The trade is exact: the number of resolvable elements in one order is Δλ_FSR/Δλₘᵢₙ = (λ/m)/(λ/mN) = N, whatever m is. A higher order shares the same N elements over a narrower span, finer but shorter; echelle gratings push this to m ≈ 50–100 with a second disperser to unstack the orders.

mN assumes every groove is lit, identical, and exactly in place. Illumination first: R = W sin θ/λ counts the width the beam covers, so a 12 mm beam on a 30 mm grating gets 12 mm worth of grooves and the ruled width is irrelevant. Groove position errors next: an error ε in one groove shifts its phasor by 2π m ε/d in order m, so random errors of rms σ leave only a fraction e^{(−(2π m σ/d)²)} of the peak intensity — 94% for σ = d/50 in second order — and scatter the rest into a floor between orders that hides faint lines beside bright ones. Periodic errors from the ruling engine are worse: a sinusoidal wobble of amplitude ε produces ghost lines beside every true one with relative intensity (π m ε/d)², about 10⁻³ for ε = d/100 in first order, which a spectroscopist can mistake for a real line. Finally the model is scalar and far-field: it predicts line shape and position but not how much light each order gets, which the groove profile (the blaze) decides and, once grooves approach λ, needs vector theory. mN is a design ceiling; the entrance slit usually sets what you actually get.