tisdag 18 mars 2014

Blackbody as Linear High Gain Amplifier

                                      A blackbody acts as a high gain linear (black) amplifier.

The analysis on Computational Blackbody Radiation (with book) shows that a radiating body can be seen as a linear high gain amplifier with a high-frequency cut-off scaling with noise temperature, modeled by a wave equation with small damping, which after Fourier decomposition in space takes the form of a damped linear oscillator for each wave frequency $\nu$:
  • $\ddot u_\nu +\nu^2u_\nu - \gamma\dddot u_\nu = f_\nu$,
where $u_\nu(t)$ is oscillator amplitude and $f_\nu (t)$ signal amplitude of wave frequency $\nu$ with $t$ time, the dot indicates differentiation with respect to $t$, and $\gamma$ is a small constant satisfying $\gamma\nu^2 << 1$ and the frequency is subject to a cut-off of the form $\nu < \frac{T_\nu}{h}$, where 
  • $T_\nu =\overline{\dot u_\nu^2}\equiv\int_I \dot u_\nu^2(t)\, dt$, 
is the (noise) temperature of frequency of $\nu$, $I$ a unit time interval and $h$ is a constant representing a level of finite precision.  

The analysis shows under an assumption of near resonance, the following basic relation in stationary state:
  •   $\gamma\overline{\ddot u_\nu^2} \approx \overline{f_\nu^2}$,
as a consequence of small damping guiding $u_\nu (t)$ so that  $\dot u_\nu(t)$ is out of phase with $f_\nu(t)$ and thus "pumps" the system little. The result is that the signal $f_\nu (t)$ is balanced to major part by the oscillator
  • $\ddot u_\nu +\nu^2u_\nu$, 
 and to minor part by the damping
  • $ - \gamma\dddot u_\nu$,
  • $\gamma^2\overline{\dddot u_\nu^2} \approx \gamma\nu^2 \gamma\overline{\ddot u_\nu^2}\approx\gamma\nu^2\overline{f_\nu^2} <<\overline{f_\nu^2}$. 
This means that the blackbody can be viewed to act as an amplifier radiating the signal $f_\nu$ under the small input $-\gamma \dddot u_\nu$, thus with a high gain. The high frequency cut-off then gives a requirement on the temperature $T_\nu$, referred to as noise temperature, to achieve high gain.   

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