Specific detectivity explained

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 ^{* = (\sqrt{A \Delta f)/()} }, 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 ⋅ \sqrt{Hz}/ W ) in honor of Robert Clark Jones who originally defined it.^{[1] [2]} ak{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 = (S _{n})/(ak{R) }, it is common to see the specific detectivity expressed as D ^{* = (ak{R)/(⋅ \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 ^{*} = (q λη)/(\left[ hc) (4 kT)/(R _{0} A) +2 q ^{2} η \Phi ^{-1 / 2} _{b\right]}

With q as the electronic charge,

λ 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, R _{0A} is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), η is the quantum efficiency of the device, and \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

\Delta f directly from the integration time constant t _{c} .\Delta f = (1)/(2 t _{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.Signal _{avg} = (1)/(N) ( \sum ^{N} _{i} Signal _{i} ) Noise _{rms} = \sqrt{ (1)/(N) \sum ^{N} _{i} ( Signal _{i} - Signal _{avg} ) ^{2}}

Now, the computation of the radiance

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 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.

R = (Signal _{avg})/(H G) = (Signal _{avg})/(\int dH dA _{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/cm² of emitting areaG is the total integrated etendue between the emitting source and detector surfaceA _{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.

NEP = (Noise _{rms})/(R) = (Noise _{rms})/(Signal _{avg}) H G

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 ^{*} = (\sqrt{\Delta f A _{d})/() } = \frac \frac

Notes and 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),)