# 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 ${\displaystyle D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}}$, where ${\displaystyle A}$ is the area of the photosensitive region of the detector, ${\displaystyle \Delta f}$ is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units (${\displaystyle cm\cdot {\sqrt {Hz}}/W}$) in honor of Robert Clark Jones who originally defined it.[1][2]

Given that noise-equivalent power can be expressed as a function of the responsivity ${\displaystyle {\mathfrak {R}}}$ (in units of ${\displaystyle A/W}$ or ${\displaystyle V/W}$) and the noise spectral density ${\displaystyle S_{n}}$ (in units of ${\displaystyle A/Hz^{1/2}}$ or ${\displaystyle V/Hz^{1/2}}$) as ${\displaystyle NEP={\frac {S_{n}}{\mathfrak {R}}}}$, it is common to see the specific detectivity expressed as ${\displaystyle D^{*}={\frac {{\mathfrak {R}}\cdot {\sqrt {A}}}{S_{n}}}}$.

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.

${\displaystyle D^{*}={\frac {q\lambda \eta }{hc}}\left[{\frac {4kT}{R_{0}A}}+2q^{2}\eta \Phi _{b}\right]^{-1/2}}$

With q as the electronic charge, ${\displaystyle \lambda }$ is the wavelength of interest, h is Planck's constant, c is the speed of light, k is Boltzmann's constant, T is the temperature of the detector, ${\displaystyle R_{0}A}$ is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), ${\displaystyle \eta }$ is the quantum efficiency of the device, and ${\displaystyle \Phi _{b}}$ is the total flux of the source (often a blackbody) in photons/sec/cm².

## 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 ${\displaystyle \Delta f}$  directly from the integration time constant ${\displaystyle t_{c}}$ .

${\displaystyle \Delta f={\frac {1}{2t_{c}}}}$

Next, an average signal and rms noise needs to be measured from a set of ${\displaystyle N}$  frames. This is done either directly by the instrument, or done as post-processing.

${\displaystyle {\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big (}\sum _{i}^{N}{\text{Signal}}_{i}^{2}{\big )}}$

${\displaystyle {\text{Noise}}_{\text{rms}}={\sqrt {{\frac {1}{N}}\sum _{i}^{N}({\text{Signal}}_{i}-{\text{Signal}}_{\text{avg}})^{2}}}}$

Now, the computation of the radiance ${\displaystyle H}$  in W/sr/cm² must be computed where cm² 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/cm² is known at the detector. If this is unknown, it can be estimated using the black-body radiation equation, detector active area ${\displaystyle A_{d}}$  and the etendue. This ultimately converts the outgoing radiance of the black body in W/sr/cm² of emitting area into one of W observed on the detector.

The broad-band responsivity, is then just the signal weighted by this wattage.

${\displaystyle R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}}}$

Where,

• ${\displaystyle R}$  is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
• ${\displaystyle H}$  is the outgoing radiance from the black body (or light source) in W/sr/cm² of emitting area
• ${\displaystyle G}$  is the total integrated etendue between the emitting source and detector surface
• ${\displaystyle A_{d}}$  is the detector area
• ${\displaystyle \Omega _{BB}}$  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.

${\displaystyle {\text{NEP}}={\frac {{\text{Noise}}_{\text{rms}}}{R}}={\frac {{\text{Noise}}_{\text{rms}}}{{\text{Signal}}_{\text{avg}}}}HG}$

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.

${\displaystyle D^{*}={\frac {\sqrt {\Delta fA_{d}}}{\text{NEP}}}={\frac {\sqrt {\Delta fA_{d}}}{HG}}{\frac {{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}}}$

## References

1. ^ R. C. Jones, "Quantum efficiency of photoconductors," Proc. IRIS 2, 9 (1957)
2. ^ R. C. Jones, "Proposal of the detectivity D** for detectors limited by radiation noise," J. Opt. Soc. Am. 50, 1058 (1960), doi:10.1364/JOSA.50.001058)