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 ∗ = A f N E P , where A is the area of the photosensitive region of the detector and f is the frequency bandwidth. It is commonly expressed in Jones units ( c m ⋅ H z / 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 R (in units of A / W or V / W ) and the noise spectral density S n (in units of A / H z 1 / 2 or V / H z 1 / 2 ) as N E P = S n R , it's common to see the specific detectivity expressed as D ∗ = R ⋅ 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 λ η h c [ 4 k T R 0 A + 2 q 2 η Φ b ] − 1 / 2
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 0 A 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 Φ b is the total flux of the source (often a blackbody) in photons/sec/cm².
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 wavelet will be integrated over a given time constant over a given number of frames.
In detail, we compute the bandwidth Δ f directly from the integration time constant t c .
Δ f = 1 2 t c
Next, an rms signal and noise needs to be measured from a set of N frames. This is done either directly by the instrument, or done as post-processing.
S i g n a l r m s = 1 N ( ∑ i N S i g n a l i 2 )
N o i s e r m s = σ 2 = 1 N ∑ i N ( S i g n a l i − S i g n a l a v g ) 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 = S i g n a l r m s H G = S i g n a l ∫ d H d A d d Ω B B
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 area G is the total integrated etendue between the emitting source and detector surface A d is the detector area Ω B B 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.
N E P = N o i s e r m s R = N o i s e r m s S i g n a l r m s 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 ∗ = Δ f A d N E P = Δ f A d H G S i g n a l r m s N o i s e r m s
