Parabolic Dish Antenna

How a parabolic dish works

A radio source far away — a satellite 36,000 km up, a quasar billions of light-years out — sends waves that arrive as a flat sheet, a plane wave. By the time it reaches a dish a few meters across, every part of that wavefront is essentially parallel. The problem: a plane wave is spread thin. A single small antenna intercepts almost none of it. You want to gather a big patch of that wavefront and pile all its energy onto one tiny receiver. That is exactly what a paraboloid does.

The trick is geometric. A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Spin that parabola about its axis and you get a paraboloid of revolution — the dish surface. Now send a plane wave straight down the axis. Each ray reflects off the surface, and because of the equal-path-length property, every reflected ray reaches the focus having traveled the exact same total distance. A ray that hit the rim went a long way to the surface and a short way to the focus; a ray near the vertex went a short way down and a long way to the focus. The differences cancel. All rays arrive at the focus in phase and add up coherently. Put a small feed antenna there and it collects the energy of the whole aperture.

The same dish works in reverse for transmit — antennas are reciprocal. Feed a spherical wave from the focus, and the reflector turns it into a flat, collimated wavefront leaving the dish: a tight beam. Receiving and transmitting are the same geometry run backward, which is why the same dish does both.

Paraboloid surface (axis along z, vertex at origin):

  z = (x² + y²) / (4f)            f = focal length

Focal length from depth d and diameter D:

  f = D² / (16 d)

Equal-path-length property (why it focuses):
  distance(ray to surface) + distance(surface to focus) = constant
  for every ray parallel to the axis.

Gain: why a bigger dish wins twice

The headline number for any dish is gain — how much more power it concentrates in its main direction compared to an isotropic radiator (one that radiates equally everywhere). For an aperture antenna the gain is set by how many wavelengths wide the opening is:

G = η · (π D / λ)²          (linear)
G_dBi = 10·log10[ η·(πD/λ)² ]

  D  = dish diameter
  λ  = wavelength = c / f
  η  = aperture efficiency (0.55 to 0.70 typical)

Worked example — a 0.6 m Ku-band TV dish at 12 GHz:
  λ = c/f = 3e8 / 12e9 = 0.025 m
  πD/λ = π·0.6/0.025 = 75.4
  (πD/λ)² = 5,685
  G = 0.6 · 5,685 = 3,411 linear = 35.3 dBi

Two levers control gain, and both are squared. Diameter: double D and gain goes up 6 dB (a factor of four in power). Frequency: shorter λ means more wavelengths across the same dish, so doubling the frequency also adds 6 dB. This is why higher bands (Ku, Ka) deliver more gain per meter of dish than C-band — and why deep-space networks push to X-band and Ka-band. Aperture efficiency η bundles every imperfection: feed spillover past the rim, illumination taper (the edge is deliberately dimmer than the center), strut and feed blockage, surface roughness, and ohmic loss. Real dishes land around 0.55–0.65; a very clean Cassegrain with a shaped subreflector can reach 0.70+.

Beamwidth and pointing

Gain and beamwidth are two sides of one coin: concentrating power into one direction necessarily narrows the beam. The half-power (−3 dB) beamwidth of a uniformly-illuminated circular aperture is approximately

θ_3dB ≈ 70 · (λ / D)   degrees      (≈ 58–72 depending on taper)

  0.6 m TV dish at 12 GHz:   θ ≈ 70·0.025/0.6 ≈ 2.9°
  3 m VSAT at 14 GHz:        θ ≈ 70·0.021/3   ≈ 0.5°
  70 m DSN dish at 8.4 GHz:  θ ≈ 70·0.036/70  ≈ 0.036°  (~2 arcmin)

A 2.9° beam forgives sloppy aiming — get the dish within a degree of the satellite and you are fine, which is why home installers can eyeball it with a signal-strength meter. A 0.036° beam is roughly a tenth of the Moon's apparent diameter; it must track a moving spacecraft to arcsecond precision, and the structure's thermal expansion in sunlight measurably walks the boresight. The bigger and higher-frequency the dish, the sharper the beam — and the more unforgiving the mount, encoders, and structural stiffness have to be.

Design tradeoffs: f/D, illumination, blockage

The single most important shape parameter is the f/D ratio — focal length over diameter. It decides how deep the dish is and, from the feed's point of view, how wide an angle the rim subtends:

• Low f/D (0.3–0.4): a deep dish. The rim wraps far around the focus, subtending a wide angle (an f/D of 0.35 makes the rim sit about 36° off the focus axis, a full subtended angle near 71°). It needs a feed with a broad pattern to illuminate the whole surface. Compact, but harder to feed cleanly.
• High f/D (0.5–0.7): a shallow dish. The rim subtends a narrow angle, so it needs a directive feed (a longer horn). The feed sits far out front on a long boom — more structure, more blockage, more wind load.

The feed's job is to illuminate the dish to about −10 to −12 dB at the edge. Light it too uniformly and energy spills past the rim (spillover loss, and the spillover picks up warm ground noise on receive); taper it too hard and the outer aperture is wasted (illumination/taper loss). The two losses trade against each other, and the efficiency peak is the compromise — typically an edge taper near −11 dB. On a prime-focus dish the feed and its support struts sit squarely in the beam, casting a shadow and scattering energy into the sidelobes (aperture blockage), which both lowers gain and raises the sidelobe floor. Offset and Cassegrain geometries exist largely to dodge this.

Reflector geometries compared

Geometry	Where the feed sits	Blockage	Pros	Cons	Typical use
Prime focus	At the paraboloid focus, in front of dish	High (feed + struts in beam)	Simplest, cheapest, broadband	Blockage loss, long feed line, ground-noise pickup	Amateur radio, small radar, low-cost dishes
Offset feed	Below the beam, fed off-axis slice	None (feed out of aperture)	No blockage, low sidelobes, sheds snow/rain	Asymmetric pattern, harder to point intuitively	Home satellite TV, VSAT, DBS
Cassegrain	Behind the dish; convex hyperbolic subreflector	Moderate (subreflector)	Short feed line, heavy/cryo electronics mount behind	Subreflector blockage, more parts to align	Large ground stations, radio telescopes, DSN
Gregorian	Behind the dish; concave ellipsoidal subreflector	Moderate	Real secondary focus, good for shaped optics	Longer than Cassegrain for same focal length	ALMA, some radio telescopes, big earth stations

Real systems and numbers

System	Diameter	Band / freq	Approx. gain	Beamwidth	Notes
Home DBS satellite TV (offset)	0.45–0.6 m	Ku, ~12 GHz	33–35 dBi	~3°	Offset-fed, LNB at focus, ~0.5 dB rain margin matters
VSAT terminal	1.2–2.4 m	Ku/Ka, 12–30 GHz	40–48 dBi	0.5–1°	Two-way data; tight pointing to avoid adjacent-satellite interference
Microwave backhaul link	0.3–1.8 m	6–80 GHz	35–50 dBi	0.3–2°	Shrouded for low sidelobes; line-of-sight only
NASA DSN (Goldstone, Madrid, Canberra)	34 m & 70 m	X/Ka, 8.4 & 32 GHz	72–80+ dBi	0.02–0.04°	Cassegrain, cryogenic LNA, beam-waveguide feed, ~70% efficiency
Green Bank Telescope	100 m	0.3–116 GHz	up to ~80 dBi	arcseconds	Offset, unblocked aperture, active surface with ~2,000 actuators
Arecibo (retired 2020)	305 m (fixed spherical)	up to ~10 GHz	~70 dBi	arcminutes	Spherical, not parabolic — steered by moving the feed, not the dish

The DSN's 70 m dishes show the whole story compressed: at X-band their ~74 dBi gain lets them pull in a signal from Voyager — now over 24 billion km away, radiating about 22 W into space — at a received power on the order of 10⁻¹⁸ W. That is only possible because the dish concentrates a 70 m wavefront onto a cryogenically-cooled feed and the beam is a few hundredths of a degree wide, tracking the spacecraft across the sky.

Surface accuracy: the Ruze limit

A dish can be the perfect shape on paper and still perform poorly if its real surface is rough. Random surface errors scatter energy out of the focus. John Ruze's 1966 analysis gives the efficiency penalty:

η_surface = exp[ −(4π ε / λ)² ]          ε = RMS surface error

  Gain loss (dB) = 685.8 · (ε/λ)²

Rules of thumb:  ε = λ/16 → ~2.7 dB loss (loose workmanship bar)
                 ε = λ/25 → ~1.1 dB loss (a sensible design target)
                 ε = λ/50 → ~0.3 dB loss (high-performance)

For ≈ 1 dB loss you need ε ≲ λ/25 ≈ 0.04 λ:
  12 GHz  (λ = 25 mm):   ε ≤ ~1.0 mm   → stamped steel is fine
  32 GHz  (λ = 9.4 mm):  ε ≤ ~0.38 mm  → machined panels
  100 GHz (λ = 3 mm):    ε ≤ ~0.12 mm  → precision panels + active surface
  230 GHz (λ = 1.3 mm):  ε ≤ ~0.05 mm  → ALMA-class, ~25 µm RMS panels

This is why millimeter-wave dishes are engineering marvels: ALMA's 12 m antennas hold a surface accuracy near 20–25 µm RMS — about a quarter the thickness of a human hair across a 12 m structure — and use carbon-fiber backup structures to stay stable as the Sun heats one side. The Green Bank Telescope and the DSN big dishes carry active surfaces: hundreds to thousands of actuators behind the panels that bend the dish back into shape as gravity sags it differently at every elevation angle (a fixed dish can only be perfect at one pointing — this is the "homologous design" idea, where the structure deforms into another paraboloid rather than into garbage).

When to use a parabolic dish

• You need very high gain in one direction toward a fixed or slowly-moving target — a satellite, a tower, a spacecraft. Dishes routinely exceed 40 dBi where a patch or Yagi tops out near 15–20 dBi.
• The frequency is high enough (roughly above 1 GHz) that a usefully-large dish is still physically reasonable. At HF a dish would have to be enormous; there you use wire antennas instead.
• You can mechanically point it — and keep it pointed. Dishes are inherently single-beam; steering means moving metal (or, increasingly, handing the job to a phased array).
• Bandwidth is broad. The reflector itself is frequency-independent; only the feed limits bandwidth, so one dish can serve many bands at once (the DSN feeds X and Ka simultaneously).

Choose something else when you need to steer the beam fast or in many directions at once (a phased array — Starlink user terminals dropped dishes for exactly this reason), when the target is so close that you don't need the gain, or when the wavelength is so long that the dish would be impractically huge.

Common misconceptions and pitfalls

• "The dish receives the signal." The dish receives nothing — it is a mirror. The feed at the focus is the actual antenna; the reflector just gathers and concentrates the wavefront onto it. Block the feed and the dish is deaf.
• "My offset dish is aimed too low." Offset dishes look like they point at the horizon but actually receive from much higher in the sky — the feed sits below the beam and the effective aperture is a tilted slice. This trips up first-time installers constantly.
• "A spherical dish is the same as a parabolic one." Only a parabola focuses a plane wave to a point. A sphere has spherical aberration — off-axis rays miss the focus. Arecibo was spherical on purpose (so it could steer by moving the feed, not the 305 m dish) and paid for it with a complex line-feed/Gregorian corrector.
• "Bigger is always better." A bigger dish has a sharper beam, so it demands a stiffer, more accurate, more expensive mount and tighter pointing. Past a point, surface accuracy and structural deflection — not size — set the usable gain.
• "Mesh dishes leak." A mesh reflector works fine as long as the holes are small compared to the wavelength (roughly hole ≤ λ/10). A wire-mesh dish is perfectly reflective at 1.4 GHz (λ = 21 cm) yet transparent — and useless — at 30 GHz.
• "Rain just attenuates the signal." At Ku/Ka band, rain both attenuates and adds thermal noise (warm raindrops emit), and a wet or snow-filled dish detunes the feed illumination. Link budgets carry a rain-fade margin (often several dB) precisely because the dish's clear-sky gain isn't the whole story.