How things work: an overview of
With a nod to Tove Jansson
How many photons do you
have in your mode?
Lost in translation
Point spread function
Fabry Perot Antenna
Photon noise Local oscillator
Baffle Secondary Mirror
Adaptive optics Sidelobes
Are these fundamentally different techniques, or just
different words for the same things?
Optical vs radio
• In the optical we do most of the signal
processing (imaging, spectroscopy etc)
before the detector.
• In the radio we do most of the signal
processing (imaging, spectroscopy etc) after
Blackbody radiation 1
This section is based closely on the NRAO Astr534 course, and
uses several diagrams from that course.
First, we derive an expression for blackbody radiation in
the classical limit — otherwise known as the Rayleigh-
We start by calculating the number of modes of radiation
at a given frequency that can exist in a cavity.
BTW, what is a blackbody?
Blackbody radiation 2
Standing waves (all different
wavelengths) between two
Standing waves (all same
wavelength, ie “modes”) in a two
Here, nx = 3; ny = 2.
In three dimensions, the
permitted frequencies are:
Blackbody radiation 3
An x-y plane in “n” space.
Permitted standing wave
modes are represented by
We calculate the density of
modes in this “n” space.
(BTW, we can also use antenna theory to calculate the number of
modes propagating through an optical system. It is simply
Number of modes ≈ AW/l2,
where A is beam area, W = solid angle, and l = wavelength.)
Blackbody radiation 4
Blackbody radiation 5
Blackbody radiation 6
Planck’s Law 1
Planck’s Law 2
This is the brightness (in watts) per unit frequency interval Dn,
surface area A, and solid angle W.
Planck’s Law 3
nmax ≈ 59 GHz . T (K)
We can also integrate Bn over all frequencies to
obtain the Stefan-Boltzmann law:
Mode occupation number
The mode occupation number, or mean number of photons per
mode, is given by:
For hn/kT >> 1, n is < 1. The photons behave independently and
obey Poisson statistics. This is the usual situation in optical
For hn/kT << 1, n is >> 1. The photons do not behave independently;
they obey Bose-Einstein statistics. Detect one, and that’s all the
information you need. If you’ve seen one, you’ve seen them all. This
is the usual situation in radio astronomy.
The Planck function revisited
• The number of modes is ≈ AW/l2 ≈ AWn2/c2
• The mode occupation number, or mean number of photons per
mode, is given by:
• Each photon carries energy hn
• There are 2 polarisations
• The brightness Bn of a blackbody is thus 2 x (number of modes) x
(number of photons per mode) x (energy per photon) =
per unit solid angle and surface area
At 500 nm (600 THz) and 5000 K (star):
n ≈ 0.003
In fact, if you are observing a star of angular size 1 milli-
arcseconds with a detector pixel subtending 0.1
arcseconds on the sky, n is effectively 3 x 10-7.
The photons behave independently and obey Poisson
statistics, producing “photon noise” (also known as “shot
If the only light the detector sees is coming from the star,
then the signal/noise ratio for any observation is simply:
S/N = √n,
where n is the number of photons detected during the
(Assuming, of course, a perfect detector that produces
no excess noise. With modern CCDs, the trick is to
integrate long enough that the photon shot noise
swamps the detector readout noise, which is typically a
few electrons rms.)
Much the same, except now almost all of the photons are
coming from the background (sky, telescope, instrument).
The noise is given by the square root of the number of all
of these photons detected per measurement interval.
The sensitivity is usually described by the NEP (Noise-
Equivalent Power); ie, that signal power required to give a
S/N of 1 in one second.
Let h, the quantum efficiency of the detector, be the
fraction of incident photons it actually detects (h < 1).
If the detector generates no noise of its own, it is said to be
background limited (ie, all the noise comes from the shot noise of
the background, not from the detector). In this case,
NEPBLIP = (2hnBPb/h)1/2 watts per √Hz,
n is the observing frequency
B is the post-detection bandwidth
Pb is the background power
h is the detector quantum efficiency
“BLIP” stands for “Background-Limited Infrared Performance”
all assuming that Pb >> Ps and hn >> kT.
The signal/noise ratio of an observation is just:
S/N = (Psignal/NEPBLIP). t1/2
Where t is the integration time.
If the detector does generate noise of its own, it can be
ascribed a value for its NEP, say NEPDetector
S/N = (Psignal/NEPDetector). t1/2
although to complicate things, the NEP of the detector
probably varies with the background anyway.
For hn/kT << 1, n is >> 1. This is the usual situation in
Eg, at 5 cm (6 GHz) and 10,000 K (HII region);
n ≈ 3 x 104
The photons do not behave independently; they obey
At 6 GHz, even for cool sources (2.7 K), n ≈ 9.
(At mm and sub-mm wavelengths, however, hn can
start to approach kT, where T is the temperature of the
Because we are in the Rayleigh-Jeans regime, power
is proportional to temperature:
Radio astronomers thus speak of the brightness
temperature of a source, or the antenna temperature
or the receiver temperature.
The detection process
• Optical astronomy:
– Collect photons at a CCD pixel until you have enough, like
catching rain drops in a bucket.
– The detection process destroys all the phase information (eg,
– On one pixel you can detect as many modes as you like — just
increase the field of view
• Radio astronomy:
– Measure the amplitude and phase of the radiation field.
– With one receiver you can only detect one mode; ie, you are
always diffraction limited. (Or more accurately, only receiving an
amount of signal equivalent to a diffraction-limited beam).
– Alternatively you can use direct detection; eg, a bolometer, and
have an arbitrary field of view.
Radio sensitivity limits
We describe the sensitivity in terms of a system
temperature, Tsys, made up of:
• Receiver “temperature”
• “Sky” “temperature”
• Various losses
The fundamental limit is quantum noise, ie, n = 1, or
Tquantum = hn/k = 48K/THz. However, real instruments
never approach this in the cm bands.
(For example, at 10 GHz, Tquantum ≈ 0.5 K, and Tsys is
typically 50 K.)
IF and “Back end”
• Digital Filter Bank
• Power detector/integrator
In a radio telescope, it is usual to shove the waves down a feed
horn, then convert them into an electrical current on a wire. This
process is sensitive to only one mode and one polarisation of
Now we have an electrical signal, it is “mixed” with a local oscillator
signal. The resulting IF (Intermediate Frequency) is given by
fsignal = fLO ± fIF
Image: James Di Francesco
National Research Council of Canada
For example, we might mix a 115 GHz signal with a 110 GHz local
oscillator to create a 5 GHz IF signal. This IF signal retains all the
amplitude and phase information of the original signal, but is now at
a much easier frequency to process.
There is no noise penalty in doing this, as long as we are dealing
with system temperatures, Tsys, of
Tsys > hn/k (≈ 0.5 K at 10 GHz)
The process of amplifying the signal is equivalent (in terms of noise
penalty) to heterodyning.
In both cases, we are increasing the mode occupation number by 1.
Why? Ask Heisenberg.
• With our signal now in electrical form, and converted down to a
user-friendly frequency, we can do amazing things with it. For
– We can have almost unlimited spectral resolution,
– We can simultaneously have as many spectral channels as we
– We can correlate the signal from one antenna with the signals
from as many other antennas as we like,
– We can build the SKA!
• There is no reason not to add a second receiver to the antenna, to
detect the other polarisation.
• While we’re at it, we may as well add additional receivers, each
seeing its own single spatial mode on the sky (ie, a multibeam
• Actually, we can do even better with a Phased Array Feed (PAF).
So, back to the optical…
We always use direct detection (for example, a CCD).
Why can’t we use amplifiers and heterodyne techniques
on an optical (or infrared) telescope?
Well we could, but…
We’d incur a noise penalty of one photon per mode. At
500 nm, this would be equivalent to increasing the sky
Tsky = 42,000 K
Hardly what you’d call dark time…
• Optical astronomers must ignore the phase of their photons, and
process the light before it is detected.
• Referring back to the Planck function, the energy in the signal is
proportional to the area-solid angle product (AW) of the beam.
• In fact, AW/l2 ≈ Nmodes, the number of modes.
• AW must be conserved throughout the instrument, so
instruments that accept a lot of modes (large primary mirror,
poor spatial resolution) become enormous.
• However, an instrument that operates with a single mode (ie, a
diffraction-limited beam) is the same size regardless of the size
of the telescope (8-inch Celestron to ELT). Hence the
importance of adaptive optics on ELTs.
• Because the light must be processed optically (no digital filter
banks!), achieving high spectral resolution also involves building
large pieces of hardware.
hn/kT ≈ 1
Is it better to use radio techniques or optical?
That depends in exquisite detail on the observation to
be conducted, and the technology available.
For example, CCAT (Can’t Compete with an Antarctic
Telescope) will have several spectrometers:
- Long slit echelle grating, R~1000 at 350 mm
- Parallel plate grating cavity, R~300 at 850 mm
- Heterodyne focal plane arrays, R~100,000
The perfect telescope
• Wavelength coverage: 300 nm - 30 metres
• Field of View: 2p steradians
• Integration time: days to months
The instrument as a filter
• Spatial filtering
• Spectral filtering
• Temporal filtering
• Multiplex advantage(s)
• Sensitivity is (preferably) set by fundamental limits
– Photon statistics s/n = √(no. of photons) or
– Quantum limit Tsys = hn/k
• May need to trade off resolution against sensitivity
• May need to compromise anyway (eg, seeing)
Image: FIRI team
How do we achieve spatial
– Rarely at the diffraction limit
– More usually seeing limited (lots of modes)
– Adaptive optics
– Aperture masking
– Interferometry is hard (-ish)
– Always at the diffraction limit (single mode!)
– Interferometry is easy (-ish)
How do we achieve spectral
– Diffraction grating
– Fabry Perot
– Fourier Transform Spectrometer (FTS)
– Big instruments
– Digital autocorrelator
– Digital filter bank
– Nifty electronics
• Rarely do we approach the truly fundamental limits
• Most often, we are limited by systematics, such as
– Fluctuating sky noise
– 1/f noise ( a subject in itself…)
• We deal with these by chopping, beam switching,
dark frames, calibration lamps and noise diodes,
In general, a dish will work at any frequency lower than
its design frequency.
Image: AAT Board Gillespie, White & Watt, 1979
Maybe the two tribes aren’t so
different after all.
With more than a nod to Tove Jansson