# Specific detectivity

Specific detectivity, or D*, for a photodetector is a figure of merit used to characterize performance, equal to the reciprocal of noise-equivalent power (NEP), normalized per square root of the sensor's area and frequency bandwidth (reciprocal of twice the integration time).
Specific detectivity is given by $D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}$, where $A$ is the area of the photosensitive region of the detector, $\Delta f$ is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units ($cm\cdot {\sqrt {Hz}}/W$) in honor of Robert Clark Jones who originally defined it.
Given that noise-equivalent power can be expressed as a function of the responsivity ${\mathfrak {R}}$ (in units of $A/W$ or $V/W$) and the noise spectral density $S_{n}$ (in units of $A/Hz^{1/2}$ or $V/Hz^{1/2}$) as $NEP={\frac {S_{n}}{\mathfrak {R}}}$, it is common to see the specific detectivity expressed as $D^{*}={\frac {{\mathfrak {R}}\cdot {\sqrt {A}}}{S_{n}}}$.
It is often useful to express the specific detectivity in terms of relative noise levels present in the device. A common expression is given below.
    $D^{*}={\frac {q\lambda \eta }{hc}}\left[{\frac {4kT}{R_{0}A}}+2q^{2}\eta \Phi _{b}\right]^{-1/2}$
With q as the electronic charge, $\lambda$ is the wavelength of interest, h is the Planck constant, c is the speed of light, k is the Boltzmann constant, T is the temperature of the detector, $R_{0}A$ is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), $\eta$ is the quantum efficiency of the device, and $\Phi _{b}$ is the total flux of the source (often a blackbody) in photons/sec/cm$^2$.

## 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 $\Delta f$ directly from the integration time constant $t_{c}$.
    $\Delta f={\frac {1}{2t_{c}}}$
Next, an average signal and rms noise needs to be measured from a set of $N$ frames. This is done either directly by the instrument, or done as post-processing.
    ${\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big\sum _{i}^{N}{\text{Signal}}_{i}{\big )}$
    ${\text{Noise}}_{\text{rms}}={\sqrt {{\frac {1}{N}}\sum _{i}^{N}({\text{Signal}}_{i}-{\text{Signal}}_{\text{avg}})^{2}}}$
Now, the computation of the radiance $H$ in W/sr/cm$^2$ must be computed where cm$^2$ 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$^2$ is known at the detector. If this is unknown, it can be estimated using the black-body radiation equation, detector active area $A_{d}$ and the etendue. This ultimately converts the outgoing radiance of the black body in W/sr/cm$^2$ of emitting area into one of W observed on the detector.
The broad-band responsivity, is then just the signal weighted by this wattage.
    $R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}},$
where
* $R$ is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
* $H$ is the outgoing radiance from the black body (or light source) in W/sr/cm$^2$ of emitting area
* $G$ is the total integrated etendue between the emitting source and detector surface
* $A_{d}$ is the detector area
* $\Omega _{BB}$ 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.
    ${\text{NEP}}={\frac {{\text{Noise}}_{\text{rms}}}{R}}={\frac {{\text{Noise}}_{\text{rms}}}{{\text{Signal}}_{\text{avg}}}}HG$
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.
    $D^{*}={\frac {\sqrt {\Delta fA_{d}}}{\text{NEP}}}={\frac {\sqrt {\Delta fA_{d}}}{HG}}{\frac {{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}}$