Reflector antennas

● Why? For a dipole● Great for long wavelength, terrible for short● Use a reflector to collect more; a parabola is

defined as a locus of points equidistant to a fixed point and a line

● So if we have a plane wave arriving a parabolic reflector will add all the waves arriving along its axis in phase at that point

Ae=

2

4

   

For an line normal to the axis

● to have focus P1+Q

1 = P

1+Q

2 = P

3+Q

3

   

A diagram

● total path top focal point must not depend on radius (r)

   

In maths

● Consider the r=0 and an arbitrary r

f h=r 2 f −z 2

h−z if we remove h , add z to both sides and square this:  f z 2=r 2

f 2z 2

−2 f zf 2z2

2 f z=r 2 f 2

z2−2 f z

z=r 2

4f Which is in the form   y=a x2

   

Focal Ratio

● If the dish had a diameter D the focal ratio is f/D● For large f/D the focal support legs get

cumbersome● For small f/D the field of view gets small● A typical compromise is f/D about 0.4- 0.6● There is a focal ellipsoid around focus where we

have good focus

   

Advantages

● Ae can approximate (large) geometrical area

● Does not depend on wavelength, so you can swap receivers

● Simpler than dipole arrays

A=D2 / 4

   

Far Field

● To have the plane parallel wavefronts we assumed what is the minimum distance (R)?

Let Δ be the maximum departure from a plane wave. This occurs at the edge of the reflector. The far-field distance is defined by requiring that Δ < λ/16

R2=R−2

D2

2

   

with some assumptions

● approximations

D≫ (so diffraction is small) , /2≪D2 /8

R≈ D2

8=2D2

   

Examples

● a 100m dish at 1cm gives 2000km (must be outside low earth orbit)

● a 12m dish at 21cm gives 1.3km (not very far!)

   

Beam Pattern

● Once we are in the far field, any emission from a parabola will look like a plane wave, so we can treat it as one.

● We can use Huygens' principle for wave propagation

● At long distances we can use the Fraunhofer approximation

   

Linear aperture

   

Aperture

● If we assume a variation in field across our aperture g(x) and wave length λ we get a field df from an element dx (from Huygens)

● If we see from a long way off (r >> D)df ∝g x

exp−i2 r / r

dx

r≈R x sin

 and     1r≈

1R

  so we can define    l≡sin

   

Aperture...

df ∝g x exp−i 2 R/exp −i 2 x l /dx

 so if we integrate across our aperture  and define u≡x

f l ∝ ∫aperture

gu e−i 2 l u du

● This is just a Fourier transform● As it true of transmission it is also so for

reception - reciprocity● We can use this for holography on our dish

surfaces

   

Holography setup

   

Aperture and Beam pattern

● So the far­field, the electric­field pattern of an aperture antenna is the Fourier transform of the electric field illuminating the aperture. For a simple one dimensional example

   

'Top Hat' functions

   

The 2 D case

● If we have a uniform 2D illumination we get an Airy Disk sky illumination

● Hard to deconvolve

   

Taper

● Real radio telescopes, with feeds and circular apertures, have power patterns that can be approximated by a 2­Dimensional Gaussian function (with a cutoff at the dish edge!)

● The Fourier transform of a wide gaussian is a narrow gaussian, so we get a nice smooth beam on the sky

● But we lose resolution● We usually get 

HPBW≈1.2D

   

Reflector accuracy

● Near perfect large paraboloids cost big bucks!● Real dishes have

– manufacturing errors

– gravitational distortion (bend with weight)

– wind and thermal distortion

● Ruze equation for surface efficiency versus rms accuracy:

s=exp [−4

2

]

   

Typical limit

● If the surface rms is too big we lose too much efficiency

● You can use mesh provided that the gaps are small ( < λ/16)

≈min16

corresponds to surface efficiency s≈0.54

   

Different optics

● Prime focus

   

Prime focus

● Simple design● Symmetry gives good polarization characteristics

on-axis● Refections can be a problem

– can put a lump at the dish apex

– remove feed support (C-BASS)

● can get spillover past the dish edge– This leads to ground pickup

Avoid at short wavelengths

   

Cassegrain

● Hyperbolic subreflector - focus above dish

   

Cassegrain optics

   

Cassegrain

● Reflector effectively multiplies f/D● Clever use can optimize the taper for high

efficiency and low sidelobes● Spillover sees sky (3K)● Tilt to use different receivers● Receivers easier to access

   

... BUT

● More complex (not simple paraboloid)

● Subreflector must be > 6λ to avoid diffraction problems

● Needs huge feed horns for long wavelengths● Can have major standing wave problems

   

Gregorian

   

Gregorian optics

   

Offset Gregorian

   

Offset Gregorian

   

Gregorian

● Use ellipsoidal subreflector to focus -complex surfaces

● Easier to use with an offset geometry to remove blockage (high efficiency and low sidelobes!)

● For some geometries we can retain good polarization properties (low astigmatism) even with an offset

● Can keep feed horn small

   

Others

● Ceduna radio telescope ,and JCMT sub-mm, use Naysmith optics

– stable platform for cryogenics and receivers

   

Mount types

● Hour Angle Declination (like HartRAO 26m)– simple calculations to track a source

– big counterweights

– Few made after 1970s

– Sky is does not rotate as we follow it

● Altitude Azimuth (Az-El)– more complex calculations (so what!)

– simpler mechanical design - can go BIG

– Sky rotation can be a problem● ASKAP de-rotates the whole dish on a 3rd axis