Multiple Slits & Diffraction Gratings

Each slit sends a wave towards the same distant point, and because the slits are equally spaced by d, each wave lags its neighbour by the same path difference d sin θ. In phase that is φ = 2π d sin θ / λ. The resultant is a sum of N phasors of equal length A₁, each turned by the same angle φ from the last — a chain of equal links bending through a constant angle, so the links lie on a circle and the resultant is its chord. The geometry gives A = A₁ sin(Nφ/2) / sin(φ/2), and squaring gives I = I₁ [sin(Nφ/2) / sin(φ/2)]². Test it at N = 2, where it collapses to 4 I₁ cos²(φ/2), the double-slit result you already have. Everything a grating does to light is a property of this one ratio, so read the sections that follow as the places where that ratio is large, zero, or moving.

When φ is a whole multiple of 2π the chain straightens out: every slit contributes in step, the amplitude is N A₁ and the intensity is N² I₁. These are the principal maxima, and φ = 2πm rearranges to d sin θ = mλ, with the integer m called the order. Two features of this equation matter. First, N is absent, so adding slits cannot shift a line by any angle at all — only d, λ and m place it. Second, sin θ ≤ 1 caps the order at m ≤ d/λ. That is the cap the 600 lines per millimetre grating above runs into at 633 nm: it stops at m = 2, and that second order lies at sin θ = 0.760, θ = 49.4°. If d were smaller than λ, nothing but the straight-through order would survive, which is why gratings for visible light are ruled at hundreds of lines per millimetre rather than tens of thousands.

Between two principal maxima the numerator sin(Nφ/2) vanishes N − 1 times while the denominator does not, giving N − 1 exact zeros and, between them, N − 2 weak secondary maxima that fade as N grows. The zeros sit at d sin θ = (p/N)λ for every integer p that N does not divide, so the first zero past order m lies only λ/(N d) further along in sin θ. In angle, the half-width of a principal maximum is Δθ = λ / (N d cos θₘ), shrinking as 1/N. Put numbers on it: 600 lines per millimetre, a beam covering 5.00 mm so that N = 3000 lines, first order at 589 nm. Then θ = 20.7° and Δθ = 1.26 × 10⁻⁴ rad, about 26 arcseconds. The energy bookkeeping agrees: each peak is N² times brighter but 1/N as wide, so the light in one order grows as N, exactly as N slits passing N times the power must.

How far apart do two nearby wavelengths land? Hold the order fixed and differentiate d sin θ = mλ, which gives d cos θ dθ = m dλ, so the angular dispersion is D = dθ/dλ = m / (d cos θₘ), in radians per metre of wavelength. Two wavelengths a gap Δλ apart arrive Δθ ≈ D Δλ apart. For the 600 lines per millimetre grating in first order at 589 nm, D = 1 / (1.667 × 10⁻⁶ m × 0.9355) = 6.41 × 10⁵ rad m⁻¹, so the two sodium D lines, 0.59 nm apart, are pushed 3.8 × 10⁻⁴ rad or 0.022° apart. Notice what D does and does not contain: it grows with the order m, with fine ruling 1/d, and with cos θ shrinking far out in the pattern — but N is nowhere in it. Illuminating more lines spreads the spectrum not at all.

A separation means nothing until it is compared with the width of what is being separated. The Rayleigh criterion calls two wavelengths just resolved when the principal maximum of one falls on the first zero of the other: D Δλ = λ / (N d cos θₘ). Substitute D = m / (d cos θₘ) and the geometry cancels completely, leaving R = λ/Δλ = m N. The 3000-line patch above has R = 3000 in first order, while splitting the sodium doublet needs only R = (589.0 nm)/(0.59 nm) = 998, so those lines stand 3.0 half-widths apart and separate comfortably. The N here is the number of lines the beam actually covers, not the number ruled on the plate: mask the beam down to 1.0 mm and N falls to 600, R falls to 600, and the doublet closes into one line while every angle in the spectrum stays exactly where it was.

Real slits have a width a, and one slit of width a sends nothing in the directions where a sin θ = pλ. The pattern of the grating is the product of that envelope with the N-slit factor: I = I₀ [sin β / β]² [sin(Nφ/2) / sin(φ/2)]², where β = π a sin θ / λ. A principal maximum survives only where the envelope is non-zero, and the two conditions collide when mλ/d = pλ/a, that is m = p (d/a). So an order dies only when some whole p makes p (d/a) a whole number, which needs d/a to be a ratio of whole numbers: rule d = 4a and orders 4, 8 and 12 are absent however many slits you add, because zero times N² is still zero, while d = 2.5a costs you orders 5, 10 and 15 and nothing below them. The envelope also ranks the brightnesses. With d/a = 5 it leaves order 1 at 88% of the central value and order 6 at 2.4%, which is why high orders disperse well and look faint.