The Background Radiation

In much of the optical the background blackbody radiation and the noise associated with it plays only a minor role and other forms of noise, such as shot noise, are dominant in systems. However, the SMM/THz is immersed in the long wavelength tail of the 300 K black body radiation.

Many years ago Lewis considered this issue in some detail [1]. The energy density between $\nu$ and $d\nu$ is given by

\begin{equation} \rho(\nu) = \frac{8\pi h\nu^3}{c^3}\left(\frac{1}{e^{h\nu/kT}-1}\right)d\nu \label{eq1} \end{equation}

In the long wavelength limit $h\nu << kT $, Eq. \eqref{eq1} can be simplified to [2]

\begin{equation}\rho(\nu) = \frac{8\pi kT\nu^3}{c^3} d\nu \end{equation}

with average energy per mode

\begin{equation}\left\langle E \right\rangle = kT\end{equation}

and number of photons per mode

\begin{equation}n_m = \frac{kT}{h\nu}\end{equation}

The power emitted in bandwidth $d\nu$ into a solid angle $\alpha$ in a direction $\theta$ relative to the normal of an emitting area $A$ is

\begin{equation} P = \frac{8 \pi kT \nu^2d\nu}{c^3}\frac{cA}{4}\frac{\alpha}{2\pi}\cos\theta \end{equation}

For a single diffraction limited mode,

\begin{equation} A\alpha\cos\theta \approx \left(\frac{\lambda}{2}\right)^2\end{equation}

giving the familiar relation

\begin{equation} P = kT d\nu \label{eq7}\end{equation}

For the power within a typical 1 MHz spectroscopic line width, at 300 K, P = 4.14 x 10^-15 W.

In the long wavelength limit of the SMM/THz, source temperature is a useful concept. Solving Eq. \eqref{eq7} for $T$ yields the temperature required of a blackbody source to produce a power $P$ in bandwidth $\Delta\nu$

\begin{equation} T= \frac{P}{k\Delta\nu} \label{eq8}\end{equation}

This leads directly to the concept of source brightness, in units of W/Hz. Source brightness is often a better figure of merit than source power. In terms of interactions with physical systems what matters is not the total available power, but rather the power within the system bandwidth, because only this power interacts with the physical system. Moreover, the power outside this linewidth is often detrimental because it can add to overall system noise.