What a thousand slits buy that two cannot

A grating is a screen with many identical openings in it, evenly spaced, and the pattern behind it is the two-slit calculation with more terms in the sum. Adding terms moves no bright direction at all. It narrows every one of them, in proportion to the number of terms, and that is the whole of what a grating is for.

Everything below is where those numbers come from, or what they cost.

The sum over N slits, and what it does not contain #

The construction is the one behind every diffraction pattern: every point of a wavefront is a fresh source, and the contributions arriving in a given direction add with their phases. For a grating the sources come in a row, so light leaving neighbouring openings a distance $d$ apart towards an angle $\theta$ differs in path by $d\sin\theta$ — the same amount for every neighbouring pair.

That makes the total a geometric series rather than an integral. Writing $u = \pi d\sin\theta/\lambda$ for half the phase step, the sum of $N$ unit contributions has magnitude $\sin Nu/\sin u$, and the intensity, normalised to one at its peaks, is

$$I(\theta) = \left[\frac{\sin Nu}{N \sin u}\right]^2.$$

The principal maxima are where the denominator vanishes and the whole expression rises to one: $u = m\pi$, which is

$$d\sin\theta = m\lambda.$$

There is no $N$ in that equation: it is the statement two openings make and the statement a million make. Two sources already fix every bright direction the grating will ever have.

What the count does change shows up by drawing the same grating at the smallest count there is, and then at a thousand.

The width goes as $1/N$, and the figures check that rather than asserting it: $N$ times the measured width comes out at 12.692°, 12.691° and 12.691° at 64, 256 and 1024 slits, and 12.705° at the twenty of the hero — a tenth of a per cent above the asymptote already. Two slits sit 12.9 per cent above it, at 14.333°, for a reason worth keeping: at $N = 2$ the expression collapses to $\cos^2 u$ exactly, whose half-maximum is at $u = \pi/4$, and no asymptotic argument applies to it.

Between the principal maxima sit $N-2$ secondary ones. At twenty slits the tallest value the curve reaches away from an order is 4.8 per cent of a peak, close to the large-$N$ limit of 4.7 per cent — a fraction that does not improve with $N$. What improves is that the secondary maxima crowd into what reads as a dark field while the principal maxima shrink to hairlines.

From a width to a resolving power #

Two nearby wavelengths through one grating give patterns whose maxima sit at slightly different angles, so resolution becomes width against displacement, as for two point sources through a lens.

The displacement is arithmetic: differentiating the grating equation at fixed order, $d\,\Delta(\sin\theta) = m\,\Delta\lambda$. The width is the figure’s business, and the first zero of the pattern lies at $\Delta(\sin\theta) = \lambda/(Nd)$. Rayleigh’s convention puts the peak of one line on the first zero of the other, and equating the two expressions gives

$$R = \frac{\lambda}{\Delta\lambda} = mN.$$

The order, times the number of lines the beam actually covers. Nothing about the ruling’s quality, the blank’s size, the material or the polish survives into that expression.

One step further is worth taking, because the result is not what a specification sheet suggests. If the illuminated width is $W$, then $N = W/d$, and the order at the working angle is $m = d\sin\theta/\lambda$. Multiplying,

$$R = \frac{W\sin\theta}{\lambda}.$$

The pitch has cancelled. Resolving power depends on how wide the beam is and what angle the light leaves at, and not on how finely the surface was ruled: halving $d$ doubles $N$ at a given order, but it also moves that order to twice the sine of the angle, so the gain belongs to the angle. At a fixed working angle — which is what a mounted spectrograph has — halving $d$ changes nothing whatever. The ceiling follows, since $\sin\theta \le 1$: $R \le W/\lambda$, the illuminated width in wavelengths, or 84,800 for a 50 mm beam at 589 nm, whatever is scratched on the glass.

That reciprocity is the most surprising thing on this page. The width of the beam is doing two apparently unrelated jobs with the same number: it sets how finely two directions can be told apart, and how finely two colours can. Both are statements about how many wavelengths fit across the illuminated region, which is the only length in the problem.

The sodium doublet, resolved on the drawing #

The standard demonstration is a pair of lines famous for being nearly the same. Sodium’s D lines sit at 589.0 and 589.6 nm, a separation of 0.6 nm out of 589, so telling them apart needs $R = 982$ — in first order, 982 illuminated lines.

The dip at exactly $R = \lambda/\Delta\lambda$ is computed rather than quoted: 18.9 per cent. That is worth setting beside the Airy pattern’s 26.5 per cent at its own Rayleigh separation — one convention applied to differently shaped peaks, and therefore two different visibilities. Neither is a threshold in the physics: a dip appears once the peaks are 0.83 of the Rayleigh separation apart, which is the inflection of one sitting on the summit of the other.

Two panels reporting the same number from different hardware is the strongest form the claim takes: the drawn sums, measured independently, agree.

How many orders there are, and which one to use #

Orders are not free. One exists only if there is an angle for it, and $\sin\theta = m\lambda/d$ cannot exceed one, so $m \le d/\lambda$: a coarse grating has many orders and a fine one few, which is the opposite of the intuition that finer ruling means more of everything.

Two consequences follow, both the instrument-maker’s. The first is overlap: order $m$ at $\lambda$ and order $m+1$ at $\lambda m/(m+1)$ arrive at the same angle, so the range an order carries without collision — its free spectral range — is $\lambda/m$. First order is free over more than an octave; order sixty at 589 nm is free over 9.8 nm and nothing more. A high order buys resolving power and takes the spectral range away in the same proportion, which is why an echelle spectrograph, working at orders of thirty to a hundred, must sort its overlapping orders with a second dispersing element crossed against the first. That element is very often a prism: the device that displaced the prism from spectroscopy cannot manage its own output without one.

The second is that the zeroth order is where the light wants to go. In the pattern above $m = 0$ is the straight-through beam, as bright as any other order and carrying no dispersion whatever, since all wavelengths arrive at the same angle. A grating of the kind drawn here throws a large fraction of its light into that useless direction. The fix is to abandon the flat screen and cut the surface into tilted facets, each mirror-reflecting the beam towards a chosen order: the blaze. The facet angle does not alter $d\sin\theta = m\lambda$ and so moves no maximum, but it moves the envelope — the single-facet diffraction pattern that multiplies the whole grating sum — so that its broad central lobe lands on the order the designer wants. A well-blazed grating delivers 60 to 80 per cent of the incident light into one order, and that number, rather than anything about resolution, is the difference between the physics and an instrument.

What it costs #

A grating spectrograph is specified by a number of lines illuminated, not by a precision. The entry that matters is lines per millimetre times the beam width, which the collimator sets. A superbly ruled grating underfilled by the beam performs like the small grating the beam covers, and stopping the instrument down loses resolution in proportion to the aperture lost.

Resolution and light compete for the entrance slit. A spectrograph’s resolution is the worse of the grating’s $mN$ and the width of the entrance slit’s image, and narrowing the slit to reach the grating’s limit throws light away in proportion. Every practical instrument sits where the two are matched.

Ghosts are a defect no count repairs. A ruling engine driven by a screw imparts the screw’s periodic error to the line positions, and a periodic modulation of a periodic structure throws satellite lines out beside every real one — Rowland ghosts, at $10^{-4}$ to $10^{-6}$ of the parent. They are indistinguishable in form from faint real lines, they scale with the parent’s brightness, and more lines multiply them too. Gratings recorded holographically, as the interference pattern of two laser beams on photoresist, have no periodic ruling error at all, at the cost of more scattered light.

The prism lost twice over, and once on arithmetic alone. A prism disperses through the variation of refractive index with wavelength, and its resolving power is the base length times $\mathrm{d}n/\mathrm{d}\lambda$ — a few thousand for a 50 mm block of dense flint, improvable only with a more dispersive glass or more prisms. The same 50 mm illuminated on a grating has a ceiling of 84,800, and modern echelles deliver $10^5$ routinely. The second defeat was decisive earlier: a grating’s dispersion, $\mathrm{d}\theta/\mathrm{d}\lambda = m/(d\cos\theta)$, is calculable in advance from a ruling pitch with no material data in it whatever, whereas a prism’s must be measured for the particular glass. That is why a grating turns wavelength into an absolute measurement — an angle and a pitch — while prism spectroscopy could only compare one line against another.

Where the model stops #

The derivation assumes a plane, monochromatic wave at normal incidence covering $N$ identical openings uniformly. A real spectrograph violates every clause.

Illumination is not normal, and the equation grows a term. With the beam arriving at $\theta_i$ the condition becomes $d(\sin\theta_i + \sin\theta_m) = m\lambda$, which is what makes the Littrow mounting — incidence and diffraction along one line, $2d\sin\theta = m\lambda$ — standard. The resolving-power argument does not change; every angle in the figures does.

Illumination is not uniform, and the count is therefore soft. A real beam dies towards its edges, so the outer lines contribute less than the inner ones and the effective count falls short of the geometric one, typically by 10 to 20 per cent. The tapering also fills in the zeros: an apodised sum has no exact nulls, so the peak of one line never sits on a true zero of the other and the Rayleigh construction becomes an approximation to itself.

The $1/N$ law is asymptotic, and the figures say by how much. At two slits the measured $N \times$ FWHM is 14.333° against the limiting 12.691°, 12.9 per cent wide; by twenty slits the discrepancy is 0.11 per cent. The law is safe for anything anybody would call a grating, and wrong by a noticeable margin for the double slit the whole argument is built on.

The slits are treated as points, and they are not. The sum gives every opening the same amplitude in every direction, which holds only if each is far narrower than a wavelength. A real opening has its own diffraction pattern, and it multiplies the grating sum as an envelope.

And the whole treatment is scalar, which a blazed grating is not. Modelling light as one oscillating quantity throws away the direction of the oscillation. Grooves with depths comparable to a wavelength diffract the two polarisations with efficiencies that can differ by more than a factor of two, and near the Rayleigh–Wood anomalies — angles at which a neighbouring order emerges at grazing — efficiency changes abruptly in a way scalar theory has nothing at all to say about. It is computed by solving Maxwell’s equations at the groove profile, numerically.

The same arithmetic somewhere else entirely #

$N$ interfering contributions and a width going as $1/N$ is not an optical result but a result about summing a geometric series, and the same expression governs instruments that share no hardware.

A phased array of radio antennas is a grating run in reverse: $N$ elements spaced $d$ apart, transmitting, the beam pointed by giving each a programmed phase rather than by turning anything. Its beamwidth is $\lambda/(Nd)$, so resolution is bought by element count — and when $d$ exceeds half a wavelength the higher orders of $d\sin\theta = m\lambda$ appear as grating lobes, the multiple orders above met again as a nuisance.

X-ray diffraction from a crystal is the same sum with the lattice as the ruling. Bragg’s condition $2d\sin\theta = m\lambda$ is the grating equation with a reflection’s factor of two in it, and the reason a crystal produces reflections of extraordinary sharpness is that $N$ is the number of illuminated lattice planes — $10^4$ to $10^6$. That the diffracting wave can be matter rather than light changes only which wavelength appears.

An acousto-optic deflector is a grating made of sound: a travelling acoustic wave modulates a crystal’s refractive index periodically, and light crossing it meets a grating whose pitch is the acoustic wavelength and which changes electronically in microseconds. The number of directions it can steer between is the number of acoustic periods across the optical beam — the same $N$ — which the engineering literature calls the time–bandwidth product.

The unifying statement is broader still: for any interferometer, resolving power is the largest path difference the instrument produces divided by the wavelength. A grating’s is $W\sin\theta$ across the illuminated width, giving $R = W\sin\theta/\lambda$ as derived; a Fourier-transform spectrometer’s is the travel of its moving mirror, which is why its resolution is quoted as a displacement in centimetres. Bandwidth and duration trade against each other this way in every wave problem, and spectral resolution is that trade with a path difference standing in for a duration.

Rittenhouse’s hair and Rowland’s engine #

The first grating was made of hair. In 1785 David Rittenhouse, in Philadelphia, strung hairs across the threads of two fine screws to make about a hundred parallel obstacles to the inch, and reported the coloured maxima and their angles in a letter to Francis Hopkinson. The pitch was around a quarter of a millimetre; the physics was entirely right; and it was read by almost nobody.

Fraunhofer arrived at the same device independently in Munich around 1821, first with wires wound between screws and then by ruling glass with a diamond, and did the thing Rittenhouse had not: he used it to measure. The dark lines he had catalogued in the solar spectrum acquired absolute wavelengths, because a grating converts a wavelength into an angle and a pitch, both measurable with a dividing engine. Prism spectroscopy could say that two lines differed; grating spectroscopy said by how many nanometres, and quantitative spectroscopy dates from that change.

Henry Rowland turned it into engineering. From 1882 at Johns Hopkins he built ruling engines around screws corrected by a lapping process of his own devising, and produced gratings of 14,438 lines to the inch — 568 to the millimetre, close to the 566 the figures above are drawn at — over blanks six inches wide. Eighty-six thousand illuminated lines give $R = 86{,}000$ in first order, ninety times what the sodium doublet asks. He also curved them: a concave grating disperses and focuses in one element, which removed the last piece of glass from the ultraviolet, where glass does not transmit. His solar atlas and his wavelength tables were the standards for forty years.

What Rowland changed closes the argument. Once $R = mN$, spectroscopic resolution stopped being a question of optical skill and became one of how many accurate lines a machine can cut. It moved from the optician’s bench to the machine shop and has stayed there: the successors to his engines are interferometrically controlled, and the gratings in the largest telescopes are mosaics of segments phased together, no single blank being rulable wide enough.

What the picture cannot show #

Every figure here plots intensity against angle for a grating lit by a plane wave of one or two wavelengths. Three things are consequently absent.

The first is efficiency. Normalising every principal maximum to exactly one is what makes the claim about positions and widths legible, and it hides that the orders of a real grating are wildly unequal in brightness, that the blaze is the whole reason a grating is usable, and that a scalar calculation could not compute the distribution anyway. The most important number about a working grating appears on none of these axes.

The second is the coherence of the illumination. The doublet figures add intensities, correct for two atomic lines; the grating figures assume all $N$ lines are lit by one wavefront, which is what makes them interfere at all. Both assumptions are invisible here: partially coherent light across the grating’s width draws a pattern that looks like a smaller $N$, so nothing distinguishes a small grating from a large one badly illuminated. Whether two contributions can interfere is prior to every result on this page and cannot be read off a graph of it.

The third is the ruling itself. A ghost at $10^{-5}$ of its parent lies four decades below the bottom of these axes, so a grating with catastrophic periodic error draws the same curve as a perfect one at this scale. The defects that decide whether a grating is worth its price sit below the resolution of a plot whose peaks reach one.

The ladder from here #

Later rungs on this anchor: the grating equation in a general mount, with incidence and diffraction both off-normal and Littrow as its special case; grating efficiency computed from Maxwell’s equations at a real groove profile, where the blaze and the polarisation dependence stop being qualitative; the instrumental profile of a real spectrograph, apodised illumination and entrance slit together, which is what a measured line shape actually is; the same sum in three dimensions, which is how crystal structures are determined; and Fourier optics, in which the far-field pattern is the transform of the aperture and a grating is a comb, so every result above becomes a corollary of one theorem.

The neighbouring ladders are diffraction at a single opening, which is the envelope over everything drawn here; the resolution of an imaging system, which is the reciprocal quantity; coherence, which decides whether the sum may be taken at all; and dispersion by refraction, the mechanism a grating displaced. Beyond them lie the spectra gratings were built to read, where the sharpness of a line stops being an optical question and becomes an atomic one.