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 Δ f N E P {\displaystyle D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}} , where A {\displaystyle A} is the area of the photosensitive region of the detector, Δ f {\displaystyle \Delta f} is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units ( c m ⋅ H z / W {\displaystyle 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 R {\displaystyle {\mathfrak {R}}} (in units of A / W {\displaystyle A/W} or V / W {\displaystyle V/W} ) and the noise spectral density S n {\displaystyle S_{n}} (in units of A / H z 1 / 2 {\displaystyle A/Hz^{1/2}} or V / H z 1 / 2 {\displaystyle V/Hz^{1/2}} ) as N E P = S n R {\displaystyle NEP={\frac {S_{n}}{\mathfrak {R}}}} , it is common to see the specific detectivity expressed as D ∗ = R ⋅ A S n {\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.
D ∗ = q λ η h c [ 4 k T R 0 A + 2 q 2 η Φ b ] − 1 / 2 {\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, λ {\displaystyle \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 {\displaystyle R_{0}A} is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), η {\displaystyle \eta } is the quantum efficiency of the device, and Φ b {\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 Δ f {\displaystyle \Delta f} directly from the integration time constant t c {\displaystyle t_{c}} .
Δ f = 1 2 t c {\displaystyle \Delta f={\frac {1}{2t_{c}}}}Next, an average signal and rms noise needs to be measured from a set of N {\displaystyle N} frames. This is done either directly by the instrument, or done as post-processing.
Signal avg = 1 N ( ∑ i N Signal i ) {\displaystyle {\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big (}\sum _{i}^{N}{\text{Signal}}_{i}{\big )}} Noise rms = 1 N ∑ i N ( Signal i − Signal avg ) 2 {\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 H {\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 A d {\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.
R = Signal avg H G = Signal avg ∫ d H d A d d Ω B B , {\displaystyle R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}},}where
- R {\displaystyle R} is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
- H {\displaystyle H} is the outgoing radiance from the black body (or light source) in W/sr/cm2 of emitting area
- G {\displaystyle G} is the total integrated etendue between the emitting source and detector surface
- A d {\displaystyle A_{d}} is the detector area
- Ω B B {\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.
NEP = Noise rms R = Noise rms Signal avg H G {\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.
D ∗ = Δ f A d NEP = Δ f A d H G Signal avg Noise rms {\displaystyle D^{*}={\frac {\sqrt {\Delta fA_{d}}}{\text{NEP}}}={\frac {\sqrt {\Delta fA_{d}}}{HG}}{\frac {{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}}}See also
This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.
References
R. C. Jones, "Quantum efficiency of photoconductors," Proc. IRIS 2, 9 (1957) ↩
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) /wiki/Doi_(identifier) ↩