Townes Noise

Because SMM/THz frequencies are 10 – 1000 times lower than optical frequencies, emission lifetimes (which go as $\nu^3$) are relatively long and experiments and sensors ordinarily operate in absorption rather than emission. Additionally, except at cryogenic or interstellar temperatures experiments are done in the long wavelength tail of the blackbody radiation. As a result there is often an intimate interaction among the noise in sources, detectors, and backgrounds.

Absorption spectrometers operate in a very different noise regime than imagers, radars, or communications systems. They detect a small fractional change in a relatively large amount of power rather than a small amount of power against a background noise. Accordingly, we need to consider an additional, and often dominant source of noise, which we shall refer to as Townes noise \[1]. This noise arises because in the mode containing the probe power and the thermal noise power, the linear superposition is of their electric fields, not of their powers. The voltage $V$ of the probe power $P$ in a waveguide of impedance $Z$ is

\[ V^2 = 2ZP , \]

and the noise fluctuation voltage $\Delta V$ in a bandwidth $\Delta\nu$ is

\[ \left(\Delta V\right)^2 = 4ZkT\Delta\nu \]

the power in the waveguide is then

\[ \frac{\left(V \pm \Delta V\right)^2}{2Z} =P \pm 2\sqrt{2kTP\Delta\nu}+ 2kT\Delta\nu \]

As a numerical example with a 1 s integration time (which also sets the bandwidth over which noise is measured in the waveguide) and 1 mW of probe power, the cross term is of the order 10^-12 W. This noise contribution is larger than the last (and of Eq. 13) by approximately

\[ \sqrt\frac{P}{kT\Delta\nu}\]

This factor is > 10⁸ for our numerical example.