Specific detectivity

Template:Short description Template:Third-party 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 $D^{*}=\frac{\sqrt{A\mathrm{\Delta }f}}{NEP}$, where $A$ is the area of the photosensitive region of the detector, $\mathrm{\Delta }f$ is the bandwidth, and NEP the noise equivalent power (in units of $W/\sqrt{Hz}$). 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 $\mathfrak{\Re }$ (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 $NEP=\frac{S_n\sqrt{\mathrm{\Delta }f}}{\mathfrak{\Re }}$, it is common to see the specific detectivity expressed as $D^{*}=\frac{\mathfrak{\Re }\cdot \sqrt{A}}{S_n}$.

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.

$D^{*}=\frac{q\lambda \eta }{hc}{[\frac{4kT}{R_{0}A}+2q^2\eta {\mathrm{\Phi }}_b]}^{-1/2}$

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 ${\mathrm{\Phi }}_b$ is the total flux of the source (often a blackbody) in photons/s/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 $\mathrm{\Delta }f$ directly from the integration time constant $t_c$.

$\mathrm{\Delta }f=\frac{1}{2t_c}$

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.

${\text{Signal}}_{\text{avg}}=\frac{1}{N}\left(\sum _i^N{\text{Signal}}_i\right)$

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

Now, the computation of the radiance $H$ in W/sr/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.

$R=\frac{{\text{Signal}}_{\text{avg}}}{HG}=\frac{{\text{Signal}}_{\text{avg}}}{\int dHd{A}_{d}d{\mathrm{\Omega }}_{BB}},$

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 area,
• $G$ is the total integrated etendue between the emitting source and detector surface,
• $A_d$ is the detector area,
• ${\mathrm{\Omega }}_{BB}$ 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.

$D^{*}=\frac{\sqrt{\mathrm{\Delta }f{A}_d}}{\text{NEP}}=\frac{\sqrt{\mathrm{\Delta }f{A}_d}}{HG}\frac{{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}$

See also

• Sensitivity (electronics)