### 8.2 The Rayleigh-Jeans Region and thermal noise.

The ‘Rayleigh-Jeans Region’ is mentioned a number of times in these pages. This term describes the ‘long wavelength’ portion of the electromagnetic spectrum radiated by a Black Body thermal source. If we look at a suitable physics (or astronomy) book we can discover that an object that is perfectly emitting/absorbing at all frequencies will tend to radiate with a brightness given by the Planck formula.

This formula indicate the power radiated per Hz bandwidth, per unit surface area into a unit solid angle (i.e. into one Sterad). The brightness (i.e. intensity) has a peak value at a frequency given by the formula

where is the temperature of the object, and are the Planck and Boltzmann constants.

We can now define what we mean by the ‘long wavelength’ region of the spectrum by saying it is where the signal frequencies are . When this is true, we can expect that and appropriate equation 8.7 by

where is the signal wavelength.

Now the thermal radiation from a Black Body is simply ‘thermal noise’ going under another name. This statement may seem incorrect. After all, we expect thermal noise to be ‘white’ – i.e. to have a power spectral density (power per frequency interval) which does not change with frequency. Yet expression 8.9 clearly shows an intensity which is frequency dependent.

Figure 8.3 shows what happens when we actually observe a large Black Body with an antenna whose effective area is . The surface is assumed to be large enough to totally fill the field of view of the antenna. The received power spectral density will be

Hence in a small bandwidth, , we will find a power level

reaching the antenna from the Black Body surface.

This result arises because two effects counteract as we change the signal wavelength. The beam’s solid angle varies with wavelength in a way that exactly balances the change in surface brightness with wavelength. When we change to a lower frequency the surface brighness falls, but the angle increases and more surface comes into view. Hence we observe ‘white’ thermal noise where the power per bandwidth interval is uniform.

Now one of the properties of a single-mode coherent antenna is that it only responds to a specific polarisation of the input signal. (A pair of wires also has this property as it is only the voltage – i.e. the potential difference – between the wires that will be seen.) The Black Body radiates ‘unpolarised’ energy – i.e. its polarisation state varies randomly from moment to moment. Hence we find that, on average, half the radiated power would pass through a given polariser and half would be rejected. The polarisation sensitivity of a single-mode receiver therefore causes it to ignore or reject half of the available thermal power, so we only see an amount.

Hence we find that the Black Body source behaves just like a ‘white’ thermal noise source when viewed by a coherent detector/antenna in the Rayleigh-Jeans (long wavelength) region of the spectrum. This equivalence also explains a ‘puzzle’ that appears when considering the thermal noise emitted by, say, a resistor in an electronic circuit. A question that pops us occasionally goes as follows: “The thermal noise power is ‘white’ and hence is proportional to the range of frequencies. If this is true and if the resistor can emit thermal noise at all frequencies, then the total thermal noise power it produces must be infinite!” The solution to this conundrum is now clear. Since the thermal noise from a resistor is actually Planck Black Body noise it only looks ‘white’ at frequencies . At higher frequencies the intensity falls away, and hence the total noise power is finite. We can now – in the Rayleigh-Jeans region – use the temperature of a thermal source to represent its brightness as the actual power radiated is simply proportional to its temperature.

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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.