N-Slit Interference-Diffraction

Put two slits in a beam and you get the gentle cosine fringes of Young's experiment. Put six, or six hundred, or sixty thousand, and something far sharper appears: the broad fringes collapse into narrow, brilliant lines with faint ripples in the dark gaps between them. That transition — from two coherent sources to many — is the birth of the diffraction grating, and the mathematics that governs it is one of the most elegant results in physical optics, because it is literally a Fourier transform. Here is why. A row of $N$ identical slits of width $a$ and spacing $d$ is one slit convolved with a comb of $N$ points. In the Fraunhofer regime the far field is the Fourier transform of the aperture, and a transform turns convolution into multiplication — so the pattern factorizes into a smooth single-slit envelope $\mathrm{sinc}^2\beta$ times an array factor $[\sin(N\gamma)/\sin\gamma]^2$ , the transform of the comb. The width $a$ decides how bright each direction can be; the spacing $d$ decides where the sharp maxima sit. Position and brightness are governed by different lengths. The array factor is the star of the show. It peaks to a height $N^2$ wherever all $N$ phasors align — the principal maxima at $d\sin\theta=m\lambda$ — and between consecutive principals threads through $N-1$ exact zeros, leaving $N-2$ weak secondary maxima. A recurring textbook error badly overstates those secondaries; in truth the strongest is only a few percent of a principal and falls as $N$ grows, the standard large- $N$ estimate being $4/(9\pi^2)\approx4.5\%$ . Getting that number right is where this article does its most careful work.

Key takeaways

$N$ identical equally spaced slits produce a far field that factorizes as (single-slit diffraction envelope) $\times$ ( $N$ -slit interference array factor).
The array factor is $[\sin(N\gamma)/\sin\gamma]^2$ , with principal maxima of height $N^2$ where $d\sin\theta=m\lambda$ ; between adjacent principals lie exactly $N-1$ zeros and $N-2$ secondary maxima.
Principal maxima grow in height as $N^2$ while narrowing as $1/N$ , so integrated power scales only as $N$ and the pattern sharpens into grating lines with resolving power $R=mN$ .
The strongest secondary maximum is small and shrinks with $N$ : $5.7\%$ of a principal at $N=6$ , $4.8\%$ at $N=20$ , with the standard large- $N$ estimate $4/(9\pi^2)=4.5\%$ — it can never rival the principal.
Key idea: the $N$ -slit pattern is the Fourier transform of a single slit convolved with an $N$ -point comb, giving $I=I_0\,\mathrm{sinc}^2\beta\,[\sin(N\gamma)/\sin\gamma]^2$ — the width $a$ (through $\beta$ ) sets brightness, the spacing $d$ (through $\gamma$ ) sets the sharp maxima.

The core insight

The far-field intensity of $N$ identical equally spaced slits is a single-slit envelope times an $N$ -slit array factor, $I(\theta)=I_0\,\mathrm{sinc}^2\beta\,[\sin(N\gamma)/\sin\gamma]^2$ . The array factor is the transform of an $N$ -point comb: it spikes to height $N^2$ at the principal maxima $d\sin\theta=m\lambda$ and dips to zero $N-1$ times between them. Because the peaks narrow as $1/N$ while rising as $N^2$ , more slits mean sharper, brighter lines — the physics of every grating.

Where this lives today

The $N$ -slit array factor is the mathematical core of the diffraction grating, and so of nearly every spectrometer, where the $N^2$ line height and $1/N$ width make spectral lines both bright and sharp. But the same array factor $[\sin(N\gamma)/\sin\gamma]^2$ governs any set of $N$ equally spaced coherent emitters. In radio and radar it is the radiation pattern of a phased-array antenna: a controllable phase per element steers the principal maximum electronically, with no moving parts — the idea now steering optical beams in solid-state LIDAR. The condition $d\sin\theta=m\lambda$ , generalised to three dimensions, is the Laue/Bragg law of crystallography: a crystal is a vast periodic array of scatterers, and the $N^2$ sharpening with the number of coherently illuminated planes is why Bragg reflections are needle-sharp, letting X-ray and electron diffraction read out atomic structure. Acousto-optic modulators, fibre Bragg gratings, and radio-telescope arrays all trace back to the same handful of slits.

The physical picture

Picture $N$ phasors of equal length, each rotated from the previous one by the same angle $2\gamma=(2\pi d/\lambda)\sin\theta$ . When $\gamma$ is a multiple of $\pi$ every phasor points the same way and they add head-to-tail into a line of length $N$ — a principal maximum of intensity $N^2$ . Tilt the angle slightly and the phasors fan out, closing into a polygon (net zero) the moment the total wrap reaches $2\pi$ — a phase change of only $\pi/N$ . That is why the maxima are $N$ times narrower than a two-slit fringe: more phasors, smaller tilt to curl them back to zero.

Figure

The N-slit pattern is two factors at once: the smooth single-slit envelope (dashed) sets how bright each order can be, while the sharp comb of principal maxima (solid, height N²=36) sits where d·sinθ = mλ; here d = 3a, so the m = ±3 orders fall on an envelope zero and go missing

The pattern is a product of two independent physics: the spacing $d$ says where the waves add in phase (the sharp lines), while the width $a$ says how much light each slit sends in a direction (the smooth envelope). When $d$ is an integer multiple of $a$ , an envelope zero lands on a principal maximum and that order disappears — the missing orders of Grating Equation. The next figure zooms into one gap to settle a question textbooks routinely botch: how bright are the ripples between the lines?

:::svg[Zoom on one period of the N = 6 array factor [sin 6γ / sin γ]²: principal maxima reach N² = 36, with N−1 = 5 zeros and N−2 = 4 secondary maxima between them; the strongest secondary is only 5.7% of the principal (standard large-N estimate ≈ 4.5%), nowhere near the ≈ 97% a careless formula predicts]

::: For large $N$ these subsidiary features become negligible and the principals look like clean grating lines on a dark background — but the count is exact for every $N$ : always $N-1$ zeros and $N-2$ secondary maxima per gap.

The core equations

In the Fraunhofer regime — recalled from Fraunhofer Diffraction — the amplitude of $N$ identical slits is a single-slit amplitude times a coherent sum of $N$ position phases. Squaring gives the intensity as a product of two factors: the diffraction envelope from the slit width and the array factor from the periodic spacing.

I(\theta)=I_0\left(\frac{\sin\beta}{\beta}\right)^2\left[\frac{\sin(N\gamma)}{\sin\gamma}\right]^2,\qquad \beta=\frac{\pi a}{\lambda}\sin\theta,\qquad \gamma=\frac{\pi d}{\lambda}\sin\theta \qquad (1)

where: $I_0$ — intensity one slit sends forward; $\beta$ — half the phase spread across a slit of width $a$ ; $\gamma$ — half the phase step between adjacent slits of spacing $d$ (at oblique incidence use $\sin\theta-\sin\theta_i$ ); $N$ — number of coherently illuminated slits. The first bracket is the single-slit $\mathrm{sinc}^2$ envelope (zeros set by $a$ ), the second the array factor (peaks set by $d$ ). The array factor is large only where every slit adds in phase — where $\gamma$ is an integer multiple of $\pi$ . There both numerator and denominator vanish, and the l'Hôpital limit gives the coherent sum of $N$ unit phasors: amplitude $N$ , intensity $N^2$ . These are the principal maxima, and $\gamma=m\pi$ written in angle is the grating equation.

\gamma=m\pi\ \Longleftrightarrow\ d\sin\theta_m=m\lambda,\qquad \left[\frac{\sin(N\gamma)}{\sin\gamma}\right]^2_{\gamma\to m\pi}=N^2 \qquad (2)

where: $m$ — integer diffraction order; $\theta_m$ — angle of the $m$ -th principal; $N^2$ — intensity enhancement over one slit at each principal. Positions depend only on $d$ and $\lambda$ ; the height $N^2$ is the signature of $N$ sources fully in phase. Away from the principals the array factor vanishes wherever $\sin(N\gamma)=0$ but $\sin\gamma\ne0$ . Counting those zeros between two consecutive principals fixes the fine structure of the pattern, and the nearest zero to a principal sets its angular width — the origin of grating resolving power.

\gamma=\frac{p\pi}{N}\ (p\ne mN)\ \Rightarrow\ I_{\text{array}}=0;\qquad N-1\ \text{zeros},\ \ N-2\ \text{secondary maxima};\qquad \Delta\theta_{1/2}=\frac{\lambda}{Nd\cos\theta_m} \qquad (3)

where: $p$ — integer indexing the zeros; $\Delta\theta_{1/2}$ — half-width of a principal (first zero from the peak), scaling as $1/N$ since the ruled width is $Nd$ . With the Rayleigh criterion this gives the resolving power $R=\lambda/\Delta\lambda=mN$ , owned by Grating Resolving Power and Dispersion. Between those zeros sit the $N-2$ secondary maxima, whose brightness textbooks most often botch. A naive formula can make the strongest secondary nearly as tall as a principal — impossible, since off-peak the phasors largely cancel. The honest value comes from the array factor near $N\gamma=3\pi/2$ , where $\sin(N\gamma)\approx\pm1$ .

\left(\frac{I_{\text{sec}}}{I_{\text{principal}}}\right)_{\!\max}=\frac{1}{N^2}\left[\frac{\sin(N\gamma)}{\sin\gamma}\right]^2_{\,N\gamma\approx 3\pi/2}\ \xrightarrow[N\to\infty]{}\ \frac{4}{9\pi^2}=0.0450 \qquad (4)

where: $I_{\text{sec}}$ — intensity of the strongest secondary; the exact finite- $N$ maxima are $0.0572$ ( $N=6$ ) and $0.0480$ ( $N=20$ ), and the standard large- $N$ estimate, taking the secondary at $N\gamma=3\pi/2$ , is $4/(9\pi^2)=0.0450$ . The lobe shrinks to a few percent of the principal as $N$ grows; it never approaches unity. A final observation ties the whole pattern to this article's portrait. The aperture is one slit convolved with an $N$ -point comb, and the far field is its Fourier transform; since the transform turns convolution into multiplication, the pattern is forced to be (transform of one slit) $\times$ (transform of the comb) — envelope times array factor, with no approximation. That array theorem is derived in full below.

Derivation AThe array factor from a geometric series

The goal of this card is to derive the array factor $[\sin(N\gamma)/\sin\gamma]^2$ from first principles by summing the $N$ equal wavelets as a geometric series, and to read straight off it the three facts that define grating behaviour: the principal height $N^2$ , the maxima at $\gamma=m\pi$ , and the $N-1$ zeros that make peaks $N$ times narrower than a two-slit fringe. It is one elementary sum and its modulus.

Step 1. Give slit $n$ the single-slit amplitude $E_1$ times a phase $e^{i\,2n\gamma}$ , since a slit displaced by $nd$ adds a path $nd\sin\theta$ , i.e. a phase $2n\gamma=(2\pi/\lambda)nd\sin\theta$ . The total amplitude is $E=E_1\sum_{n=0}^{N-1}e^{i2n\gamma}$ . Why: in the far field the slits radiate along parallel rays, so their only relative phase is the constant step $2\gamma$ set by the spacing — the array is a chain of equal phasors turning by a fixed angle.

Step 2. Sum the geometric series with ratio $e^{i2\gamma}$ : $\sum_{n=0}^{N-1}e^{i2n\gamma}=\dfrac{e^{i2N\gamma}-1}{e^{i2\gamma}-1}=e^{i(N-1)\gamma}\,\dfrac{\sin(N\gamma)}{\sin\gamma}$ , using $e^{ix}-1=2i\sin(x/2)e^{ix/2}$ . Why: a finite sum of equal-ratio phasors is exactly summable, and the exponential prefactor is a pure phase carrying no intensity.

Step 3. Take the modulus squared: $I=|E|^2=I_0\,\mathrm{sinc}^2\beta\,[\sin(N\gamma)/\sin\gamma]^2$ ; the prefactor $e^{i(N-1)\gamma}$ has unit modulus and drops out. At $\gamma\to m\pi$ both $\sin(N\gamma)$ and $\sin\gamma$ vanish, and l'Hôpital gives the ratio $\to N$ , so the intensity there is $N^2$ times a single slit. Why: at a principal maximum every phasor points the same way and the chain straightens to length $N$ ; the $0/0$ limit is the coherent sum of $N$ unit vectors.

Step 4. Find the zeros: $\sin(N\gamma)=0$ with $\sin\gamma\ne0$ means $\gamma=p\pi/N$ for $p\ne mN$ . Between two consecutive principals ( $\gamma$ advancing by $\pi$ ) there are exactly $N-1$ such zeros and hence $N-2$ secondary maxima; the first zero sits $\pi/N$ from each principal, giving the $1/N$ width. Why: the phasor chain closes into a zero-sum polygon each time the wrap $2N\gamma$ passes a multiple of $2\pi$ without all phasors realigning — $N-1$ times per gap. *These results — the geometric-series sum, the $N^2$ peak height, the $N-1$ zeros and $N-2$ secondary maxima per gap, and the $N=1,2$ limits ( $\sin2\gamma/\sin\gamma=2\cos\gamma$ ) — all follow directly from the steps above.*

Go deeper · Derivation B · 2 minThe strongest secondary maximum, and a formula that cannot be right

Derivation BThe strongest secondary maximum, and a formula that cannot be right

This card fixes a genuine error from the source material for this article: an estimate that the first subsidiary maximum has intensity $I_{\text{sec}}\approx N^2/(1+(N/\pi)^2\pi^2)$ , whose middle expression reduces to $N^2/(1+N^2)\to1$ — a secondary lobe almost as bright as the principal, a physical impossibility. The goal here is to derive the honest value $4/(9\pi^2)$ and see why the bad formula fails, since the true strongest secondary is a few percent of the principal and decreases with $N$ .

Step 1. First, the physical veto. The array factor is $N^2$ at a principal because $N$ phasors align; a secondary maximum sits where they nearly cancel, leaving a short residual. A residual rivalling the aligned sum would contradict the geometry and violate energy bookkeeping — the principals already carry essentially all the light (peak $N^2$ , width $1/N$ , power $\propto N$ ). Any formula giving $I_{\text{sec}}\to$ principal is wrong before a number is computed. Why: the secondary lobes live in the near-cancellation region, so their amplitude is set by how badly the polygon fails to close, not by the coherent sum.

Step 2. Locate the strongest secondary. It sits in the first gap past a principal, near where $\sin(N\gamma)$ hits its extreme, $N\gamma\approx 3\pi/2$ . There $|\sin(N\gamma)|\approx1$ , so the array factor is $\approx 1/\sin^2\gamma$ . Why: the numerator saturates at $\pm1$ between zeros, so the secondary peak is controlled by the slowly varying $\sin\gamma$ .

Step 3. For large $N$ this secondary sits at small $\gamma\approx 3\pi/(2N)$ , so $\sin\gamma\approx\gamma$ and the array factor is $\approx[2N/(3\pi)]^2=4N^2/(9\pi^2)$ ; dividing by the principal $N^2$ gives $4/(9\pi^2)=0.0450$ . Why: near the axis the denominator linearizes and the $N$ -dependence cancels in the ratio, leaving a pure number.

Step 4. At finite $N$ the exact maximum, found numerically, is $0.0572$ at $N=6$ and $0.0480$ at $N=20$ , descending as $N$ grows toward the few-percent level; the standard textbook estimate for the large- $N$ value is $4/(9\pi^2)=0.0450$ . Contrast the bad formula: at $N=6$ it gives $36/37=0.973$ , off by a factor of seventeen and qualitatively wrong in its $N$ -trend (it rises toward $1$ ). Why: the secondary lobe fades relative to the ever-taller principal as $N$ grows, the exact opposite of a formula that saturates at unity — the corrected value is what a real grating shows. *These results are confirmed by the derivation above: the exact ratios are $0.0572$ ( $N=6$ ) and $0.0480$ ( $N=20$ ), the standard large- $N$ estimate is $4/(9\pi^2)=0.0450$ , and the base formula evaluates to $0.97$ — a physically impossible value.*

Go deeper · Derivation C · 1 minThe pattern as a Fourier transform of rect convolved with a comb

Derivation CThe pattern as a Fourier transform of rect convolved with a comb

The aim of this card is to show that the factorization is not coincidence but a consequence of one theorem — the Fraunhofer far field is a Fourier transform, so the transform of a convolution is a product. Writing the $N$ -slit aperture as one slit convolved with a comb makes the envelope-times-array-factor structure inevitable, and it is why this article's portrait is the Fourier transform.

Step 1. Write the aperture as $t(x)=\mathrm{rect}(x/a)*\sum_{n=0}^{N-1}\delta(x-nd)$ : a single slit of width $a$ convolved with a comb of $N$ Dirac teeth spaced $d$ . Convolving a shape with a comb simply copies that shape to each tooth — exactly $N$ identical slits. Why: convolution with $\delta(x-nd)$ translates a function to $x=nd$ , so a comb of deltas stamps out $N$ copies of the slit at the grating positions.

Step 2. The Fraunhofer field is the Fourier transform of $t$ , evaluated at spatial frequency $u=\sin\theta/\lambda$ (recalled from Fraunhofer Diffraction): $\tilde t(u)=\mathcal F\{t\}(u)$ . Why: in the far field each direction $\theta$ samples one spatial-frequency component of the aperture, which is the definition of the transform.

Step 3. Apply the convolution theorem, $\mathcal F\{f*g\}=\mathcal F\{f\}\cdot\mathcal F\{g\}$ . The transform of the slit is $a\,\mathrm{sinc}(au)$ — the envelope, $\beta=\pi a u$ — and the transform of the $N$ -tooth comb is $\sum_n e^{-i2\pi n d u}=e^{i\phi}\sin(N\pi d u)/\sin(\pi d u)$ — the array factor, $\gamma=\pi d u$ . Why: convolution in position becomes multiplication in frequency, so the pattern splits into two independent factors set by $a$ and $d$ .

Step 4. Multiply and square: $I(\theta)\propto\mathrm{sinc}^2\beta\,[\sin(N\gamma)/\sin\gamma]^2$ , recovering equation $(1)$ . The envelope's zeros come from the slit ( $a\sin\theta=p\lambda$ ), the array's peaks from the comb ( $d\sin\theta=m\lambda$ ), and their coincidence gives the missing orders. Why: the transform pair rect $\leftrightarrow$ sinc and comb $\leftrightarrow$ array-factor are exact, so the product is the full intensity with no small-angle or large- $N$ approximation. *The convolution-theorem factorization and the transform pairs (rect $\to$ sinc, $N$ -comb $\to$ array factor) follow directly from the steps above.*

Limiting cases

The general result collapses to every familiar interference pattern.

Case	Result	Meaning
Single slit, $N=1$	$I=I_0\,\mathrm{sinc}^2\beta$	the array factor is $1$ ; only the diffraction envelope survives
Two slits, $N=2$	$I=4I_0\,\mathrm{sinc}^2\beta\,\cos^2\gamma$	using $\sin2\gamma/\sin\gamma=2\cos\gamma$ — Young's fringes under an envelope
Grating limit, $N\to\infty$	principals $\to$ delta-like lines, $R=mN$	ideal grating; secondary maxima wash into a smooth background
Missing orders, $d=qa$	$m=q,2q,3q,\dots$ extinguished	an envelope zero coincides with a principal maximum
Strongest secondary (large $N$ )	$4/(9\pi^2)=4.5\%$ of a principal	subsidiary lobes never rival the principal and shrink with $N$

Common trapThe secondary maxima between grating lines are almost as bright as the principal maxima

They are not — not remotely. This misconception is fed by a plausible-looking but wrong formula, $I_{\text{sec}}\approx N^2/(1+(N/\pi)^2\pi^2)=N^2/(1+N^2)$ , which tends to $1$ and so predicts a secondary lobe nearly matching the principal. That is impossible: the array factor is $N^2$ at a principal precisely because all $N$ phasors align, while a secondary maximum lives where the phasor polygon almost closes. The honest strongest secondary is only a few percent — $5.7\%$ at $N=6$ , $4.8\%$ at $N=20$ , standard large- $N$ estimate $4/(9\pi^2)=4.5\%$ — and it falls as slits are added, the opposite of what the bad formula claims. Sanity-check any secondary estimate against energy conservation: the principals hold height $N^2$ over width $1/N$ , so they carry essentially all the power, leaving only crumbs between.

Worked examples

1 · Counting and sizing the ripples for six slits (confidence builder)

Six identical slits are ruled with spacing $d=1666.67$ nm and illuminated at normal incidence by $\lambda=633$ nm light. Locate the first principal maximum, state its height, and find how many secondary maxima lie between principals and how bright the strongest one is.

Step 1. The first principal is the grating equation with $m=1$ : $\sin\theta_1=\lambda/d=633/1666.67=0.3798$ , so $\theta_1=\arcsin(0.3798)=\mathbf{22.32^\circ}$ — the same order angle as a full grating of this pitch, because position depends only on $d$ .

Step 2. Its height is $N^2=6^2=\mathbf{36}$ times the single-slit intensity in that direction. All six phasors align, so the amplitude is $6$ and the intensity $36$ .

Step 3. Between adjacent principals there are $N-1=\mathbf{5}$ zeros (at $\gamma=p\pi/6$ , $p=1,\dots,5$ ) and $N-2=\mathbf{4}$ secondary maxima. The strongest sits near $N\gamma=3\pi/2$ , i.e. $\gamma\approx\pi/4$ , where the array factor is $2.06$ ; dividing by $36$ gives $\mathbf{5.7\%}$ of the principal.

Step 4. Compare the discredited textbook formula $N^2/(1+N^2)=36/37=0.973$ , which would put the ripple at $97\%$ of the line — seventeen times too high and qualitatively wrong. First principal at $22.32^\circ$ , height $36\times$ ; $5$ zeros and $4$ secondary maxima per gap, the strongest just $5.7\%$ of the principal.

2 · What happens as the grating grows (exam level)

Keep the same pitch and wavelength but increase the number of illuminated slits from $6$ to $20$ to $30000$ (a $50$ mm grating at $600$ l/mm). Track the strongest secondary maximum and the width and resolving power of the first-order line.

Step 1. The strongest secondary maximum falls with $N$ : from $5.7\%$ at $N=6$ to $\mathbf{4.8\%}$ at $N=20$ , settling to a few percent, with the standard large- $N$ estimate $4/(9\pi^2)=\mathbf{4.5\%}$ . The ripples do not vanish, but they sink into a smooth, faint background relative to ever-brighter lines.

Step 2. The first-order line sharpens as $1/N$ . Its angular half-width is $\Delta\theta_{1/2}=\lambda/(Nd\cos\theta_1)$ ; with $Nd=30000\times1666.67$ nm $=50$ mm and $\cos\theta_1=0.925$ , this is $633\times10^{-9}/(0.050\times0.925)=1.37\times10^{-5}$ rad $=\mathbf{13.7\ \mu\text{rad}}$ .

Step 3. The resolving power is $R=mN=1\times30000=\mathbf{30000}$ , so at $589$ nm it separates wavelengths differing by $\Delta\lambda=\lambda/R=19.6$ pm — enough to split the sodium D-doublet ( $0.6$ nm) roughly thirty times over. Secondaries fade from $5.7\%\to4.8\%\to4.5\%$ ; the line narrows to $13.7\ \mu\text{rad}$ and resolves $R=mN=30000$ .

Physics of the portrait

The portrait of this article is not a person but an operation — the Fourier transform — because the $N$ -slit pattern is not merely described by a transform, it is one.

Origins · 1 minfourier transform and how this was discovered

fourier transform

the far field is the transform of the aperture · Fourier optics and diffraction

The Fourier transform decomposes any function into its sinusoidal components, and in the Fraunhofer regime it is the machine that turns an aperture into its diffraction pattern: each observation direction reads off one spatial-frequency component of the opening. The $N$ -slit array is the cleanest demonstration. Its aperture is a single rectangle convolved with a comb of $N$ points, and because the transform sends convolution to multiplication, the pattern is forced to be a product — a $\mathrm{sinc}$ envelope from the rectangle times an array factor from the comb. The transform's operations become physical facts: a slit and its diffraction envelope have reciprocal widths (narrow slit, broad envelope), and the more periods you illuminate the sharper the transform of the comb, which is why resolving power grows as $N$ . The mathematics descends from Joseph Fourier, who introduced his series and integral to solve the flow of heat and, in doing so, handed optics the language it now speaks natively — a lens and a distance do to light exactly what the transform does to a function.

History

1807

— Joseph Fourier presents his theory of heat to the Paris Institute, introducing the decomposition into sinusoids that becomes the Fourier transform

1821

— Joseph von Fraunhofer formulates the far-field theory of the diffraction grating

1822

— Fourier's Théorie analytique de la chaleur lays out the transform in full

1835

— Friedrich Magnus Schwerd works out the quantitative multiple-slit diffraction pattern

1874

— Lord Rayleigh derives the grating's chromatic resolving power,

R=mN

Going further

Texts. Hecht, Optics, 5th ed., §10.2 (Fraunhofer diffraction: the many-slit pattern and the grating) · Foot, Atomic Physics, §A.2 (diffraction gratings and resolving power) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 4 (Fourier optics of periodic apertures) · Goodman, Introduction to Fourier Optics, 4th ed., Ch. 4 · Demtröder, Laser Spectroscopy, Vol. 1 (grating spectrometers and resolution). Key checkpoints for the core results: (i) the geometric-series sum giving $[\sin N\gamma/\sin\gamma]^2$ ; (ii) the $N=1,2$ limits ( $\sin2\gamma/\sin\gamma=2\cos\gamma$ ); (iii) the peak value $N^2$ at $\gamma\to m\pi$ ; (iv) the first-zero half-width $\gamma=\pi/N$ and hence $\Delta\theta=\lambda/(Nd\cos\theta_m)$ ; (v) $R=mN$ by combining dispersion with the Rayleigh criterion; and (vi) missing orders when $d/a$ is integer. The illustrative numbers $R=2000$ and $0.25$ nm are standard textbook values.

Provenance addendum (correction). The source material's secondary-maximum formula, $I_{\text{sec}}\approx N^2/(1+(N/\pi)^2\pi^2)=N^2/(1+N^2)\to1$ , is physically wrong: a secondary lobe cannot approach the principal. This article corrects it to the standard result — the strongest secondary maximum is only a few percent of the principal: the exact maxima are $0.0572$ ( $N=6$ ) and $0.0480$ ( $N=20$ ), and the standard large- $N$ estimate, taking the secondary at $N\gamma=3\pi/2$ , is $4/(9\pi^2)=0.0450$ .

Prerequisites. Fraunhofer Diffraction · Single-Slit Diffraction · Huygens–Fresnel Principle · Fresnel Number. Leads to. Grating Equation · Grating Resolving Power and Dispersion · Rayleigh Resolution Criterion.

Why do the principal maxima get sharper as I add more slits?

Because the first zero moves closer to the peak. A principal is flanked by a zero a phase $\pi/N$ away in $\gamma$ , so its angular half-width is $\lambda/(Nd\cos\theta)$ — inversely proportional to the total illuminated width $Nd$ . Killing a maximum only needs the extreme slits to drift half a wavelength out of step, and with more slits that misalignment builds over a smaller tilt. So doubling the slit count halves the line width while raising the peak fourfold — why a grating with tens of thousands of lines resolves doublets a two-slit interferometer never could.

If the peak intensity is N² times a single slit, where does all that energy come from?

It is redistributed, not created. The single-slit envelope fixes how much light leaves each slit in a direction; adding slits does not change the total power, only where it lands. At a principal the $N$ contributions add in phase, giving amplitude $N$ and intensity $N^2$ — but the peak is only $1/N$ as wide, so the energy under it scales as $N^2\times(1/N)=N$ , matching the $N$ slits' share. Between principals the light is removed by destructive interference and piled into the lines. Energy is conserved; interference merely concentrates it.

What exactly was wrong with the secondary-maximum formula this article corrects?

The formula $I_{\text{sec}}\approx N^2/(1+(N/\pi)^2\pi^2)$ simplifies to $N^2/(1+N^2)\to1$ — it claims the brightest ripple is almost as tall as a grating line. That cannot happen: a principal reaches $N^2$ because all phasors align, whereas a secondary lives where they nearly cancel. The exact strongest secondary is a few percent — $5.7\%$ at $N=6$ , $4.8\%$ at $N=20$ — and the standard large- $N$ estimate is $4/(9\pi^2)=4.5\%$ , from the array factor at $N\gamma=3\pi/2$ . The trend is opposite too: the true ratio decreases with $N$ , while the bad formula rises toward $1$ .

How is this different from Young's two-slit interference?

It is the same physics with more sources, and $N=2$ is a special case. Setting $N=2$ gives $[\sin2\gamma/\sin\gamma]^2=4\cos^2\gamma$ , the familiar cosine-squared fringes under the single-slit envelope. What changes as $N$ grows is sharpness: two slits give broad, equally spaced maxima with nothing between them; many slits give narrow lines (width $\propto1/N$ ) separated by dark gaps with only faint secondary ripples. The maxima stay put — fixed by $d\sin\theta=m\lambda$ — but become far sharper and taller.

Does the single-slit envelope shift the positions of the principal maxima?

No. The envelope and array factor depend on different lengths and act independently: the principal positions are set entirely by the spacing $d$ through $d\sin\theta=m\lambda$ , while the envelope, set by the width $a$ , only scales the brightness at each fixed angle. The envelope can dim an order, or extinguish it if one of its zeros ( $a\sin\theta=p\lambda$ ) lands on a principal — the missing-order case — but it never moves a maximum. This clean split of position from brightness is exactly what the convolution/Fourier picture guarantees.

Why is the far-field pattern of N slits a Fourier transform at all?

Because in the Fraunhofer regime each direction collects wavelets that left the aperture along parallel rays, so their only relative phase is linear in position — and summing linear-phase contributions across the aperture is the definition of a Fourier transform. The aperture and its far field form a transform pair, with angle as spatial frequency $u=\sin\theta/\lambda$ . For $N$ slits the aperture is a rectangle convolved with a comb, so its transform is a product of a sinc and an array factor. It is the same reason a lens, which brings the far field to its focal plane, is called an analog Fourier-transform engine.