Editing Specific detectivity (section)

== Detectivity measurement ==
Detectivity can be measured from a suitable optical setup using known parameters.
You will need a known light source with known irradiance at a given standoff distance. The incoming light source will be chopped at a certain frequency, and then each wavelength will be integrated over a given time constant over a given number of frames.

In detail, we compute the bandwidth <math>\Delta f</math> directly from the integration time constant <math>t_c</math>.
: <math> \Delta f = \frac{1}{2 t_c} </math>

Next, an average signal and [[root mean square|rms]] noise needs to be measured from a set of <math>N</math> frames. This is done either directly by the instrument, or done as post-processing.
: <math> \text{Signal}_{\text{avg}} = \frac{1}{N}\big( \sum_i^{N} \text{Signal}_i \big) </math>
: <math> \text{Noise}_{\text{rms}} = \sqrt{\frac{1}{N}\sum_i^N (\text{Signal}_i - \text{Signal}_{\text{avg}})^2} </math>

Now, the computation of the radiance <math>H</math> in W/sr/cm<sup>2</sup> must be computed where cm<sup>2</sup> is the emitting area. Next, emitting area must be converted into a projected area and the [[solid angle]]; this product is often called the [[etendue]]. This step can be obviated by the use of a calibrated source, where the exact number of photons/s/cm<sup>2</sup> is known at the detector. If this is unknown, it can be estimated using the [[black-body radiation]] equation, detector active area <math>A_d</math> and the etendue. This ultimately converts the outgoing radiance of the black body in W/sr/cm<sup>2</sup> of emitting area into one of W observed on the detector.

The broad-band responsivity, is then just the signal weighted by this wattage.
: <math>R = \frac{\text{Signal}_{\text{avg}}}{H G} = \frac{\text{Signal}_{\text{avg}}}{\int dH dA_d d\Omega_{BB}},</math>
where
* <math>R</math> is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
* <math>H</math> is the outgoing radiance from the black body (or light source) in W/sr/cm<sup>2</sup> of emitting area
* <math>G</math> is the total integrated etendue between the emitting source and detector surface
* <math>A_d</math> is the detector area
* <math>\Omega_{BB}</math> is the solid angle of the source projected along the line connecting it to the detector surface.

From this metric noise-equivalent power can be computed by taking the noise level over the responsivity.
: <math> \text{NEP} = \frac{\text{Noise}_{\text{rms}}}{R}  = \frac{\text{Noise}_{\text{rms}}}{\text{Signal}_{\text{avg}}}H G </math>

Similarly, noise-equivalent irradiance can be computed using the responsivity in units of photons/s/W instead of in units of the signal.
Now, the detectivity is simply the noise-equivalent power normalized to the bandwidth and detector area.
: <math> D^* = \frac{\sqrt{\Delta f A_d}}{\text{NEP}} =  \frac{\sqrt{\Delta f A_d}}{H G} \frac{\text{Signal}_{\text{avg}}}{\text{Noise}_{\text{rms}}} </math>