Antenna Arrays and Directivity

A single antenna, however cleverly shaped, is stuck with a broad pattern: the center-fed wire of the previous article tops out near directivity $2.4$ before its beam disintegrates into lobes. Yet a 5G base station lands a beam on one phone in a crowd, and a radar tracks a target through a $1^\circ$ pencil of sky. The trick is never a better single element — it is many identical elements and interference. In directions where the path-length differences bring every element's field into phase, amplitudes add and intensity grows as $N^2$ ; a hair off that direction, the phasors curl up and cancel. An antenna array is a diffraction grating run in reverse: radiation instead of reception, engineered instead of accidental. Two ideas organize everything. Pattern multiplication says that for identical, identically oriented elements the total far field factorizes into the element pattern times a scalar array factor $\mathrm{AF}=\sum_n a_n e^{i\mathbf{k}\cdot\mathbf{r}_n}$ — the spatial Fourier transform of the excitation. All beam shaping, steering, and sidelobe control lives in that sum, wholly under the designer's control through positions and feed phases. Directivity quantifies the payoff: $D=4\pi U_{\max}/P_{\rm rad}$ , the concentration of power relative to an isotropic radiator, the number that enters every link budget through the Friis equation. This article builds both tools at Jackson-register honesty: the Hertzian and half-wave reference directivities computed exactly, the uniform-line array factor summed in closed form, the beamwidth and $-13$ dB sidelobe structure extracted, and the steering and grating-lobe conditions derived rather than quoted. The coherent $N^2$ enhancement itself is the multi-charge coherence theorem of Larmor Formula from the Liénard–Wiechert Fields, here spent on hardware.

Key takeaways

For $N$ identical, identically oriented elements, the far field factorizes: (element pattern) × (array factor $\mathrm{AF}=\sum_n a_ne^{i\mathbf{k}\cdot\mathbf{r}_n}$ ) — the array factor is the spatial Fourier transform of the excitation.
Directivity $D=4\pi U_{\max}/P_{\rm rad}$ is the pattern-concentration figure of merit: $D=1.5$ for a short dipole (exact), $1.64$ for a half-wave dipole (numeric integral); gain folds in efficiency, $G=e_{\rm rad}D$ .
The uniform $N$ -element line sums to the Dirichlet kernel $|\mathrm{AF}|=\big|\sin(N\psi/2)/\sin(\psi/2)\big|$ with $\psi=kd\cos\theta+\alpha$ : main beam at $\psi=0$ with intensity $N^2$ , first nulls at $\psi=\pm2\pi/N$ , first sidelobe $\to-13.3$ dB for large $N$ .
A progressive phase ramp $\alpha=-kd\cos\theta_0$ steers the beam to $\theta_0$ electronically; broadside beamwidth is $\mathrm{HPBW}\approx0.886\,\lambda/Nd$ and broadside directivity $D\approx2Nd/\lambda$ .
Grating lobes — full-strength duplicate beams — appear unless $d\,(1+|\cos\theta_0|)<\lambda$ ; full scan to end-fire demands $d<\lambda/2$ .
Key idea: coherence buys $N^2$ peak intensity from $N$ radiators, and the array factor — a Fourier transform you control — decides exactly where that peak points.

The core insight

Identical elements radiating coherently give a far field that factorizes as (element pattern) × (array factor); the array factor is the spatial Fourier transform of the excitation, so beam direction, width, and sidelobes are engineered by choosing element positions and feed phases.

Where this lives today

The phased array has become the default architecture of radio civilization. Every 5G base station panel is a planar array steering dozens of simultaneous beams by per-element phase and amplitude control; every Starlink user terminal is a flat phased array tracking satellites across the sky with zero moving parts; automotive radar at 77 GHz images traffic with MIMO arrays whose virtual apertures exceed their physical ones. Military and weather radars committed decades earlier — the electronically scanned array won because inertia-free beams can interleave search, track, and communication within milliseconds. Radio astronomy runs the same equations at continental scale: ALMA's sixty-six dishes and the VLA's twenty-seven are arrays whose "elements" are themselves high-gain antennas, with baselines up to kilometers giving beamwidths $\lambda/L$ in the milliarcsecond regime — aperture synthesis, deferred here to the astronomy track, is the art of filling in the missing spacings by Earth rotation. The Square Kilometre Array will push to hundreds of thousands of elements. At every scale the design tension is the one this article derived: spacing wide for aperture, tight for grating-free scan; excitation uniform for beamwidth, tapered for sidelobes; and always the $N^2$ of coherence paying for the hardware.

The physical picture

Stand far from a row of $N$ identical radiators and add up what arrives. Each element contributes a phasor of the same length; the only difference between elements is phase — geometric phase $kd\cos\theta$ per step from the extra path length, plus whatever feed phase $\alpha$ per step the designer dialed in. The total inter-element phase is $\psi=kd\cos\theta+\alpha$ , and the array factor is a chain of $N$ unit phasors, each rotated $\psi$ from the last. At $\psi=0$ the chain is a straight line of length $N$ : peak field $N$ , peak intensity $N^2$ . At $\psi=2\pi/N$ the chain closes into a regular polygon: exact null. Between nulls, the partially curled chain gives sidelobes. The entire lobe anatomy of an array is this one picture.

Figure

Array factor as a chain of element phasors

The $N^2$ is not magic and not free energy — it is the coherence theorem from Larmor Formula from the Liénard–Wiechert Fields, where fields from multiple coherent sources add before squaring. Total radiated power still scales as $N$ (for fixed drive per element); coherence merely redistributes it, stealing from most directions to enrich one. The bookkeeping closes exactly: a beam of peak $N^2$ and angular width $\sim1/N$ integrates to total power $\sim N$ . The same geometry explains resolution. To distinguish directions $\Delta\theta$ apart, the phase profile across the aperture $L=Nd$ must change by order $2\pi$ — giving $\Delta\theta\sim\lambda/L$ , the diffraction limit wearing its transmitting hat. Bigger arrays make narrower beams, and the computed curves below show the whole anatomy for a modest eight-element line: the $12.8^\circ$ main lobe, the $-12.8$ dB first sidelobe, the beam sliding under a phase ramp, and the catastrophic duplicate beams when the spacing exceeds the grating condition.

Figure

Computed array factor curves for an 8-element line

Figure

Phase-ramp steering and the grating-lobe rule

Directivity: the figure of merit

Before arrays can be compared, "how directional" needs a number. Define the radiation intensity $U(\theta,\phi)=r^2\langle S_r\rangle$ — time-averaged power per unit solid angle, finite as $r\to\infty$ precisely because radiation fields fall as $1/r$ . An isotropic radiator emitting the same total power would show $U_{\rm avg}=P_{\rm rad}/4\pi$ in every direction, so the natural concentration measure is the ratio at the pattern peak:

D=\frac{U_{\max}}{U_{\rm avg}}=\frac{4\pi\,U_{\max}}{P_{\rm rad}}=\frac{4\pi}{\Omega_A},\qquad \Omega_A\equiv\frac{1}{U_{\max}}\oint U\,\mathrm{d}\Omega \qquad (1)

where: $\Omega_A$ — the beam solid angle, the cone into which the antenna would pack all its power if the pattern were uniform at peak intensity. $D$ is a pure pattern property, dimensionless and $\ge1$ ; engineers quote $10\log_{10}D$ in dBi. A real antenna also loses power to conductor and dielectric heating, so the deliverable is the gain $G=e_{\rm rad}D$ with radiation efficiency $e_{\rm rad}=R_{\rm rad}/(R_{\rm rad}+R_{\rm loss})$ — the resistor divider priced concretely in the short-whip example of Center-Fed Linear Antenna. Directivity connects transmitting and receiving through the effective aperture $A_e=\lambda^2G/4\pi$ , and two antennas across a distance $R$ obey the Friis relation $P_r/P_t=G_tG_r(\lambda/4\pi R)^2$ — the equation every link budget starts from, and the reason directivity is the currency of radio engineering.

Derivation ADirectivity of the reference elements

The array formalism needs calibrated elements. We compute the directivity of the short (Hertzian) dipole exactly and of the half-wave dipole numerically — the two standard references (1.76 dBi and 2.15 dBi) against which array gain is quoted. The strategy is pure bookkeeping of equation (1): peak intensity over average intensity.

Step 1. Short dipole. Electric Dipole Radiation gives $U(\theta)=U_0\sin^2\theta$ : peak broadside, null on axis. The total power requires one classic integral,

P_{\rm rad}=\oint U\,\mathrm{d}\Omega=2\pi U_0\int_0^\pi\sin^3\theta\,\mathrm{d}\theta=2\pi U_0\cdot\frac{4}{3}=\frac{8\pi}{3}U_0

Why $\sin^3$ ? One $\sin^2\theta$ from the pattern, one $\sin\theta$ from the solid-angle measure — a pairing that recurs in every radiation integral of this chapter.

Step 2. Take the ratio.

D_{\rm short}=\frac{4\pi U_0}{(8\pi/3)U_0}=\frac{3}{2}=1.5\quad(1.76\ \mathrm{dBi})

Independent of length, current, and frequency: every electrically small electric-dipole radiator on Earth shares this number, because they all share the $\sin^2\theta$ donut.

Step 3. Half-wave dipole. The pattern is $U\propto\cos^2[(\pi/2)\cos\theta]/\sin^2\theta$ from Center-Fed Linear Antenna, whose power integral reduced to the cosine integral $\mathrm{Cin}(2\pi)=2.4377$ ; framed as equation (1) the ratio is

D_{\lambda/2}=\frac{4}{\mathrm{Cin}(2\pi)}=1.6409\quad(2.15\ \mathrm{dBi})

The slimmer donut (HPBW $78^\circ$ vs $90^\circ$ ) buys nine percent over the short dipole — element choice fine-tunes directivity; only arrays multiply it.

Step 4. The beam-solid-angle reading. $\Omega_A=4\pi/1.5=8.38$ sr for the short dipole, $7.66$ sr for the half-wave — both antennas still spray most of the sphere. Kraus's estimate $D\approx4\pi/\Theta_E\Theta_H$ (principal-plane half-power widths in radians) inverts this logic and predicts array directivities to tens of percent before any integral is done. Verified: $\int_0^\pi\sin^3\theta\,\mathrm{d}\theta=4/3$ exact; the half-wave value recomputed by direct numeric integration of the pattern, $D=1.640922$ , agreeing with $4/\mathrm{Cin}(2\pi)$ to ten digits. Values match Balanis Ch. 4 and Jackson §9.4.

The array factor

Now distribute $N$ identical elements at positions $\mathbf{r}_n$ with complex excitations $a_n$ . Identical elements radiate identical patterns $\mathbf{f}(\theta,\phi)$ , so in the far zone — where all separation vectors collapse to phase shifts along the common line of sight $\hat{\mathbf{n}}$ — the sum factorizes:

\mathbf{E}(\theta,\phi)=\mathbf{f}(\theta,\phi)\,\frac{e^{ikr}}{r}\underbrace{\sum_{n=0}^{N-1}a_n\,e^{ik\hat{\mathbf{n}}\cdot\mathbf{r}_n}}_{\mathrm{AF}(\theta,\phi)} \qquad (2)

where: $\mathrm{AF}$ — the array factor, a scalar interference sum carrying all dependence on geometry and feeding; $\mathbf{f}$ — the element pattern (for half-wave dipole elements, the $F(\theta)$ of the previous article). This is pattern multiplication: intensity = (element pattern)² × $|\mathrm{AF}|^2$ . Its jurisdiction is real but bounded — identical elements, identical orientations, and weak mutual coupling; the warning below prices the fine print. Read as mathematics, $\mathrm{AF}$ is the discrete spatial Fourier transform of the excitation set $\{a_n\}$ : everything known about Fourier pairs (shift theorems, window tapers, resolution limits) transcribes into antenna language.

Go deeper · Derivation B · 2 minArray factor of the uniform line and its lobe anatomy

Derivation BArray factor of the uniform line and its lobe anatomy

The workhorse geometry: $N$ equally spaced elements on a line ( $z_n=nd$ ), equal amplitudes, progressive feed phase $n\alpha$ . We sum the geometric series exactly, then extract the main beam, nulls, beamwidth, sidelobes, and directivity scaling — the complete anatomy visible in the computed curves above.

Step 1. Sum the phasor chain. Each step contributes $\psi=kd\cos\theta+\alpha$ , so with the geometric series

\mathrm{AF}=\sum_{n=0}^{N-1}e^{in\psi}=\frac{1-e^{iN\psi}}{1-e^{i\psi}}=e^{i(N-1)\psi/2}\,\frac{\sin(N\psi/2)}{\sin(\psi/2)}

The prefactor is a pure reference phase (it vanishes if the origin sits at the array center); the magnitude is the Dirichlet kernel, exactly the $N$ -slit grating function of optics.

Step 2. Main beam and the $N^2$ peak. As $\psi\to0$ , $\sin(N\psi/2)/\sin(\psi/2)\to N$ : field $N$ , intensity $N^2$ — the aligned chain of the phasor picture, the coherence theorem in hardware. The beam condition $\psi=0$ gives $\cos\theta_0=-\alpha/kd$ ; broadside ( $\theta_0=90^\circ$ ) for $\alpha=0$ , end-fire for $\alpha=\mp kd$ .

Step 3. Nulls and beamwidth. First nulls where the chain first closes: $N\psi/2=\pm\pi$ , i.e. $\psi=\pm2\pi/N$ . Near broadside, $\psi=kd\cos\theta\approx -kd\,\delta\theta$ for $\theta=90^\circ+\delta\theta$ , so the null-to-null width is $2\lambda/Nd$ and the half-power width evaluates to

\mathrm{HPBW}\approx 0.886\,\frac{\lambda}{Nd}\ \mathrm{rad}

Why $0.886$ ? It is where the Dirichlet kernel falls to $1/\sqrt2$ of peak — the same constant as the sinc function's half-power point, since for $N\gg1$ the kernel's main lobe is a sinc in disguise.

Step 4. Sidelobes. Between nulls the chain half-curls; the first sidelobe sits near $N\psi/2=3\pi/2$ with normalized amplitude $1/[N\sin(3\pi/2N)]\to 2/3\pi$ for large $N$ : intensity $-13.3$ dB. For $N=8$ the exact value is $-12.8$ dB — uniform excitation buys the narrowest beam at the price of stubborn sidelobes; amplitude tapering (Dolph–Chebyshev and kin) trades width for suppression, the window-function tradeoff of Fourier analysis.

Step 5. Directivity scaling. Integrating $|\mathrm{AF}_n|^2$ over the sphere for isotropic elements gives, for broadside and $d$ a multiple of $\lambda/2$ , the exact and satisfying $D=N$ ; the general long-array law is $D\approx 2Nd/\lambda$ — directivity is the array's electrical length. End-fire doubles it ( $D\approx4Nd/\lambda$ ) by hiding the beam's wide dimension along the axis. Verified: numeric integration for $N=8$ , $d=\lambda/2$ , $\alpha=0$ gives $D=8.000$ (exact result $N$ recovered), $\mathrm{HPBW}=12.80^\circ$ vs $0.886\lambda/Nd=12.69^\circ$ , first sidelobe $-12.80$ dB; large- $N$ sidelobe limit $20\log_{10}(2/3\pi)=-13.5$ dB in field, $-13.26$ dB from the exact sinc lobe — the quoted $-13.3$ dB. Matches Balanis Ch. 6.

Go deeper · Derivation C · 1 minSteering, scan loss, and grating lobes

Derivation CSteering, scan loss, and grating lobes

A phased array's defining move is steering without motion. We derive the phase ramp that points the beam, the price paid off broadside, and the spacing condition beyond which the array grows a second, full-strength beam. The visible-region idea does all the work: $\theta$ only ranges over real angles, so $\psi=kd\cos\theta+\alpha$ only sweeps a window of width $2kd$ — and what the Dirichlet kernel does inside that window is all that is physical.

Step 1. The steering ramp. Demand the main peak $\psi=0$ at chosen $\theta_0$ :

\alpha=-kd\cos\theta_0

A linear phase ramp across the aperture — the Fourier shift theorem enacted in copper. Programming $\alpha$ element by element moves the beam in microseconds; no gimbal, no inertia.

Step 2. Scan loss. Steered to $\theta_0$ , the array's projected aperture toward the beam is $Nd\sin\theta_0$ : the beam broadens as $1/\sin\theta_0$ and directivity falls roughly proportionally (element pattern worsens it further). Real phased arrays are specified to $\pm60^\circ$ of broadside precisely because the projection cost beyond that (factor 2 in aperture at $60^\circ$ ) becomes prohibitive.

Step 3. Grating lobes. The Dirichlet kernel peaks at every $\psi=2\pi m$ , not just $\psi=0$ . A second peak enters the visible window when some real $\theta$ satisfies $kd(\cos\theta-\cos\theta_0)=\pm2\pi$ , i.e. when

\frac{d}{\lambda}\ge\frac{1}{1+|\cos\theta_0|}\qquad\Longleftrightarrow\qquad d\,(1+|\cos\theta_0|)\ge\lambda

Why the worst case at end-fire? Steering toward the axis pushes the main peak to the window's edge, dragging the $m=\pm1$ replica in from the other side; full end-fire scan ( $|\cos\theta_0|\to1$ ) demands $d<\lambda/2$ — the array-domain Nyquist criterion, with grating lobes as aliases of undersampled space.

Step 4. The design corner this forces. Broadside-only systems may relax to $d<\lambda$ ; wide-scan systems pay for $d\le\lambda/2$ with more elements per aperture (cost, heat, coupling). The 28 GHz worked example below runs the numbers a 5G module actually faces. Verified: for $d=0.7\lambda$ steered to $\theta_0=45^\circ$ , the bound gives $d_{\max}=0.586\lambda<0.7\lambda$ — grating lobe present, confirmed by direct evaluation of the kernel (second full peak in the visible window); for $d=\lambda/2$ the bound permits scan to end-fire marginally. Matches Balanis §6.3 and Zangwill §20.6.

Limiting cases

Case	Result	Meaning
$N=1$	$\mathrm{AF}=1$ , $D=D_{\rm element}$	No interference; back to $1.5$ (short) or $1.64$ (half-wave) — the calibrated floor.
Broadside, $d=\lambda/2$	$D=N$ exactly; $\mathrm{HPBW}=0.886\lambda/Nd$	Each doubling of the array doubles directivity and halves beamwidth; the clean reference case.
End-fire ( $\alpha=\mp kd$ )	$D\approx4Nd/\lambda$	Beam hides its wide dimension along the axis; double broadside's directivity for the same length.
Grating onset	$d(1+\|\cos\theta_0\|)\ge\lambda$	Aliased duplicate beam at full strength; the Nyquist limit of spatial sampling.
$N\to\infty$ , fixed $L=Nd$	$\mathrm{AF}\to$ sinc; sidelobe $-13.3$ dB	Continuous uniform aperture recovered; discrete and continuous apertures share one asymptotic anatomy.

Common trapPattern multiplication is a theorem about identical elements, not a law of nature

The factorization (2) requires every element to radiate the same pattern in the same orientation with negligible mutual coupling. Real arrays violate all three at some level: neighboring elements load each other (mutual impedance shifts currents from their design values), edge elements see a different electromagnetic environment than central ones, and scan changes the coupling — the dreaded scan blindness, where at certain angles a coupled surface wave swallows the beam entirely. Sidelobe levels and feed impedances are the first casualties; the clean $-13$ dB of the Dirichlet kernel routinely degrades several dB in hardware. Use pattern multiplication as the design skeleton, then let full-wave simulation and measurement dress it. And never conflate $D$ with $G$ : directivity is geometry, gain is geometry times survival of your watts.

Worked examples

Example 1 — an eight-element broadside line, completely characterized. Eight half-wave dipoles, spaced $d=\lambda/2$ , driven uniformly in phase ( $\alpha=0$ ). Find the beamwidth, sidelobe level, and directivity with honest error bars on the estimates.

Step 1. Beamwidth. $\mathrm{HPBW}\approx0.886\lambda/Nd=0.886/4=0.2215$ rad $=12.7^\circ$ ; exact numeric solution of $|\mathrm{AF}_n|^2=1/2$ gives $12.8^\circ$ — the estimate is good to one percent already at $N=8$ .

Step 2. Sidelobes. First sidelobe exactly at $-12.8$ dB (the large- $N$ limit $-13.3$ dB approached from above) — visible in the computed blue curve earlier.

Step 3. Array directivity. For $d=\lambda/2$ broadside the exact isotropic-element result is $D_{\rm AF}=N=8$ ( $9.0$ dB). Numeric integration confirms $8.000$ .

Step 4. Include the element. Parallel half-wave dipoles multiply their own broadside concentration into the plane the array factor leaves wide: $D\approx D_{\rm AF}\times1.64=13.1$ ( $11.2$ dBi) — an estimate, since the product rule for directivities holds only when the two patterns concentrate in orthogonal planes; full integration of (element)²× $|\mathrm{AF}|^2$ refines it by a few percent.

Example 2 — a 28 GHz phased tile scanned to sixty degrees. A 5G module at $f=28$ GHz uses a linear strip with $d=\lambda/2$ and must scan $\pm60^\circ$ off broadside. Check the grating margin and price the scan.

Step 1. Geometry. $\lambda=c/f=10.71$ mm, so $d=5.35$ mm — millimeter-wave frequencies exist precisely so that half-wave-sampled arrays fit in a phone-adjacent box.

Step 2. Grating check. Sixty degrees off broadside is $\theta_0=30^\circ$ from the axis: the bound demands $d<\lambda/(1+\cos30^\circ)=0.536\lambda$ . The design's $0.5\lambda$ passes — with a $7\%$ margin, which is why $\lambda/2$ is the industry default for wide-scan tiles.

Step 3. Scan cost. At $60^\circ$ off broadside the projected aperture halves ( $\cos60^\circ=0.5$ ): the beam doubles in width and the directivity drops $\sim3$ dB — a loss the link budget must absorb at the cell edge.

Step 4. The budget itself. A 64-element tile at $d=\lambda/2$ broadside offers $D\approx2N d/\lambda=64$ ( $18.1$ dB). With $20$ dBi realized gain at both ends of a $200$ m street link, Friis gives $P_r/P_t=G_tG_r(\lambda/4\pi R)^2=1.8\times10^{-7}$ ( $-67.4$ dB): one watt transmitted, $180$ nW received — comfortably above thermal noise in the channel bandwidth, because the two arrays contributed $40$ dB of the closing argument.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Cambridge and Terling — the arithmetic of resolution

John William Strutt, third Baron Rayleigh, owned the physics of interference the way this article uses it: not as a phenomenon but as a budget. His criterion for resolving two sources — path difference of order a wavelength across the aperture — is exactly the beamwidth law $\Delta\theta\sim\lambda/L$ derived here for arrays, and his analysis of gratings established the sidelobe anatomy that the Dirichlet kernel formalizes. Rayleigh's fingerprints are everywhere in radio science that came after him: the resolution limit that forces radio astronomy toward kilometer baselines, the aperture arithmetic in every antenna textbook's directivity chapter, even the statistics of multipath fading (the Rayleigh distribution) that mobile-array engineers battle daily. Characteristically, he did all of it while running a self-funded laboratory at his family estate, measuring with wooden boxes and piano wire what others could not with brass — and winning the 1904 Nobel Prize partly for weighing the difference between atmospheric and chemical nitrogen to one part in a thousand. Precision bookkeeping of amplitude and phase: the array designer's entire craft, practiced before antennas existed.

History

1879

— Rayleigh states the diffraction criterion for resolving two sources — the beamwidth law every array obeys

1905

— Braun demonstrates directional wireless transmission with three phased antennas, sharing the 1909 Nobel Prize with Marconi

1926

— Yagi and Uda publish the parasitic director-reflector array, directivity without feed networks

1946

— Ryle and Vonberg operate the first radio-astronomy interferometer, arrays as synthetic apertures

1968

— The FPS-85 phased-array radar steers beams electronically across a sixty-degree cone with no moving parts

2019

— Massive-MIMO base stations make the phased array a consumer commodity at 5G frequencies

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §11.1–11.2 (radiation, antenna arrays) · Jackson, Classical Electrodynamics, 3rd ed., §9.1–9.5, §9.12 (multipole radiation, linear antenna, arrays) · Zangwill, Modern Electrodynamics, §20.5–20.6 (antennas and arrays) · Balanis, Antenna Theory: Analysis and Design, 4th ed., Ch. 2, 6 (directivity, linear arrays) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 (radiation)

Provenance. Equations are standard far-field antenna theory. A verifier should recompute: (1) the short-dipole directivity $D=4\pi/(8\pi/3)=3/2$ using $\int_0^\pi\sin^3\theta\,d\theta=4/3$ (Balanis Ch. 4; Jackson §9.2); (2) the Dirichlet-kernel array factor via the geometric series (Balanis Eq. 6-10; Zangwill §20.6); (3) the grating-lobe condition $d<\lambda/(1+|\cos\theta_0|)$ and steering relation $\beta=-kd\cos\theta_0$ (Balanis §6.3). Quoted numbers — half-wave dipole $D=1.64$ , first sidelobe $-13.3$ dB, broadside HPBW $\approx50.8^\circ\lambda/Nd$ , Friis and effective-aperture formulas — are standard textbook values (Balanis, Kraus). No nonstandard numbers introduced. Lane addendum (emadv): all anchors recomputed — $D_{\lambda/2}=1.640922$ numeric vs $4/\mathrm{Cin}(2\pi)$ to ten digits; $N=8$ , $d=\lambda/2$ broadside: $D=8.000$ exact, HPBW $12.80^\circ$ (formula $12.69^\circ$ ), first sidelobe $-12.80$ dB (large- $N$ $-13.26$ dB); grating audit at $d=0.7\lambda$ , $\theta_0=45^\circ$ (bound $0.586\lambda$ , lobe confirmed) and at the 28 GHz tile ( $d=\lambda/2$ vs bound $0.536\lambda$ ); Friis $-67.4$ dB at 200 m with $\lambda=10.71$ mm. The steering-phase symbol is $\alpha$ here (Balanis's $\beta$ ) to avoid collision with $v/c$ .

Builds on: Electric Dipole Radiation: Exact Fields and Zone Structure · Multipole Expansion of Radiation Fields · Center-Fed Linear Antenna · Larmor Formula from the Liénard–Wiechert Fields Leads to: Synchrotron Radiation · Vector Spherical Harmonics

Where exactly does the factor N squared come from if each element radiates the same power as before?

From adding fields, not powers. At the main-beam angle all $N$ element fields arrive in phase: total field $NE_1$ , intensity $N^2|E_1|^2$ . Away from that angle the phasors partially or fully cancel. Integrate the resulting pattern over the sphere and — for spacings that keep elements roughly uncoupled — the total is $N$ times one element's power, exactly what the transmitters supplied. Coherence is a spatial redistribution engine: it starves almost every direction to feed one. The same theorem, run for accelerated charges instead of driven dipoles, is why bunched electron beams radiate coherently below the bunch-length frequency — the Larmor-article version of this arithmetic.

Is the array factor really just a Fourier transform I can reason about with signal-processing intuition?

Yes, and the dictionary is worth memorizing. Element positions = sample times; excitations $a_n$ = sample values; observation direction (through $u=\cos\theta$ ) = frequency; array factor = the DTFT of the excitation sequence. Then: aperture length ↔ observation time (resolution $\lambda/L$ ↔ $1/T$ ); amplitude taper ↔ window function (uniform = rectangular window = $-13$ dB sidelobes; Chebyshev taper = equiripple design); steering ramp ↔ modulation shift theorem; grating lobes ↔ aliases from undersampling, with $d<\lambda/2$ as the spatial Nyquist criterion for full scan. Almost every array design technique is a windowing or sampling theorem transcribed — and array textbooks freely borrow filter-design tables for exactly this reason.

Why do wide-scan arrays insist on half-wavelength spacing when fewer, wider-spaced elements would be cheaper?

Because scanning drags the main beam across the visible window of $\psi$ and pulls the $m=\pm1$ kernel replicas toward it from the edges. The no-grating condition $d(1+|\cos\theta_0|)<\lambda$ tightens as the scan angle grows; scanning all the way to end-fire sends it to $d<\lambda/2$ . Violate it and a duplicate beam of full strength radiates into (or receives from) an unintended direction — for radar a false target and wasted power, for communications interference and a security hole. Broadside-only systems (fixed links, some radio astronomy configurations) legitimately relax toward $d\lesssim\lambda$ , buying aperture per element; the moment electronic scan is in the requirements, half-wave sampling returns.

How do Yagi antennas get array-like directivity with only one driven element?

By letting induced currents do the feeding. The reflector and director rods of a Yagi–Uda antenna are parasitic: the driven half-wave dipole's near field induces currents in them, and their lengths and spacings are tuned so those induced currents carry approximately the progressive phase of an end-fire array. The system is genuinely an array — pattern multiplication applies with the induced excitations — but the "feed network" is electromagnetic coupling, free and lossless, at the cost of narrow bandwidth and no steering (the phases are frozen into the geometry). Typical rooftop Yagis with a dozen elements reach $10$ – $15$ dBi, consistent with the end-fire scaling $D\approx4Nd/\lambda$ for their actual length.

What breaks first when I pack elements closer than half a wavelength?

Mutual coupling. Neighboring dipoles sit deep in each other's near fields; each element's driving-point impedance acquires terms from every neighbor's current (mutual impedances), so the currents you get differ from the currents you fed. First casualties: sidelobe levels drift dBs from the Dirichlet prediction, feed match degrades, and efficiency drops as circulating reactive power finds loss. The extreme failure is scan blindness — at particular scan angles the coupled array supports a surface wave that steals essentially all input power, blinding the radar at that angle. Superdirective designs (many closely packed, strongly opposed currents) are mathematically legal but practically fragile for exactly these reasons: tolerance, bandwidth, and ohmic efficiency collapse together.

Radio interferometers have huge gaps between antennas — do they not suffer catastrophic grating lobes?

They do, and they engineer around them rather than away. A sparse array's instantaneous beam (the "dirty beam") is riddled with grating and side structure exactly as the kernel predicts. Astronomy's rescues: baselines are laid out non-uniformly (randomized spacings smear coherent grating peaks into a pedestal), Earth's rotation sweeps each baseline through many effective orientations over hours (filling the spatial-frequency plane), and deconvolution algorithms (CLEAN and successors) remove the known beam structure from the image. The result — aperture synthesis — treats the array factor as a measured, correctable instrument response rather than a fixed pattern. The physics is this article's; the craft of inverting it belongs to the astronomy track.

Does adding more elements always increase gain, or is there a point of diminishing returns?

For a fixed spacing, directivity grows essentially linearly with $N$ ( $D\approx2Nd/\lambda$ broadside) as long as the feed network delivers coherent, low-loss excitation — and that clause is the binding constraint. Every doubling of $N$ doubles the feed's splitting stages, adding insertion loss that grows with array size; at some point added ohmic loss cancels added directivity and gain saturates even as $D$ climbs. Phase and amplitude errors impose a second ceiling: random element errors of rms $\sigma$ radians cap the achievable sidelobe floor and scatter power out of the main beam (gain degradation $\approx e^{-\sigma^2}$ ). Practical very-large arrays therefore go modular — subarrays with local amplification (active electronically scanned arrays), where each element or tile carries its own transmit/receive module and the loss ladder resets at every stage.

Antenna Arrays and Directivity

Where this lives today

The physical picture

Directivity: the figure of merit

The array factor

Limiting cases

Worked examples

Going further

Before this

Where this leads