IR Detectors and True-SB

In the previous post From Correct Planck Law to True-SB and False-SB I derived the following Stefan-Boltzmann Law based on the new proof of Planck's radiation law presented in Computational Blackbody Radiation in Slaying the Skydragon:

(True-SB) $R(T,T_b) =\int_{T_b}^{T}\gamma T f^2 df + \int_0^{T_b}\gamma (T - T_b) f^2df \equiv I_1 + I_2$,

where $R(T,T_b)$ is the radiance from a blackbody of temperature $T$ into a blackbody background of temperature $T_b < T$.

Here the first integral $I_1$ is the radiance from the blackbody absorbed as heat by the background above the cut-off of the background and $I_2$ the net heat absorbed below cut-off as the difference between absorbed and emitted radiation.

Let us see what (True-SB) can tell us about the possible design of IR detectors, where $T$ is the temperature of the object to be detected and $T_b$ that of the detector. There are two types of detectors to consider (see here):

1. cooled with $T_b < T$
2. un-cooled with possibly $T_b > T$.

In a cooled detector with $T_b < T$ both $I_1$ and $I_2$ are positive and $R(T,T_b)$ can

be directly recorded as heating transferred into an electric signal by a thermocouple. (For a star a large distance only $I_1$ is recorded, with the help of optical magnification of the input,

but this is not IR.)

In an un-cooled detector the heat transfer would be away from the detector if $T_b>T$, and

in principle the object could be detected from recording the cooling of the detector. But this is

a form of negative detection which may be difficult to get to work in practice.

More interesting is to explore what (True-SB) suggests for the design by heating in the case the object has lower temperature than the detector:

In this case the signal from the object does not contain frequencies above the cut-off of the detector and so $I_1=0$, but by magnifying the signal optically (by a lense) the radiation below cut-off can be changed into

• $I_2(M)= \int_0^{T_b}\gamma (MT - T_b) f^2df$

where $M>1$ is a magnification factor and $I_2(M)$ may thus become positive even if $T_b > T$.

We sum up the experience from (True-SB):

1. If the temperature of the detector is lower than the object, then direct recording of heating is possible (with $I_1>0$ and possibly also $I_2>0$ ).
2. If the temperature of the detector is higher than that of the object, then heating may be recorded after optical input magnification (with $I_2>0$).

Note added: I made a mistake believing that a lens can magnify radiance, which I correct in Learning by Seeing.