How things work: an overview of astronomical instrumentation. John Storey. With a nod to Tove Jansson. Or, How many photons do you have in your mode?. Lost in translation. Autocorrelator. Point spread function. Beam profile. HEMT. Fabry Perot. Antenna. Closure phase. CCD. Grating.
With a nod to Tove Jansson
Point spread function
Are these fundamentally different techniques, or just different words for the same things?
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-Jeans Law.
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?
Standing waves (all different wavelengths) between two boundaries.
Standing waves (all same wavelength, ie “modes”) in a two-dimensional cavity.
Here, nx = 3; ny = 2.
In three dimensions, the permitted frequencies are:
An x-y plane in “n” space. Permitted standing wave modes are represented by dots.
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 ≈ A/2,
where A is beam area, = solid angle, and = wavelength.)
This is the brightness (in watts) per unit frequency interval , surface area A, and solid angle .
max ≈ 59 GHz . T (K)
We can also integrate B over all frequencies to obtain the Stefan-Boltzmann law:
The mode occupation number, or mean number of photons per mode, is given by:
For h/kT >> 1, n is < 1. The photons behave independently and obey Poisson statistics. This is the usual situation in opticalastronomy.
For h/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.
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 noise”.)
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 observation.
(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 , the quantum efficiency of the detector, be the fraction of incident photons it actually detects ( < 1).
“BLIP” stands for “Background-Limited Infrared Performance”
all assuming that Pb >> Ps and h >> 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 h/kT << 1, n is >> 1. This is the usual situation in radio astronomy.
Eg, at 5 cm (6 GHz) and 10,000 K (HII region);
n ≈ 3 x 104
The photons do not behave independently; they obey Bose-Einstein statistics.
At 6 GHz, even for cool sources (2.7 K), n ≈ 9.
(At mm and sub-mm wavelengths, however, h can start to approach kT, where T is the temperature of the background. )
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.
We describe the sensitivity in terms of a system temperature, Tsys, made up of:
The fundamental limit is quantum noise, ie, n = 1, or
Tquantum = h/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.)
National Research Council of Canada
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 radiation.
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
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 > h/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.
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 temperature to:
Tsky = 42,000 K
Hardly what you’d call dark time…
In general, a dish will work at any frequency lower than its design frequency.
Image: AAT Board
Gillespie, White & Watt, 1979
With more than a nod to Tove Jansson