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 $$D^*=\frac{\sqrt{A \Delta f}}{NEP}$$, 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 ($$cm \cdot \sqrt{Hz}/ W$$) in honor of Robert Clark Jones who originally defined it.

Given that noise-equivalent power can be expressed as a function of the responsivity $$\mathfrak{R}$$ (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 $$NEP=\frac{S_n}{\mathfrak{R}}$$, it is common to see the specific detectivity expressed as $$D^*=\frac{\mathfrak{R}\cdot\sqrt{A}}{S_n}$$.

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.


 * $$D^* = \frac{q\lambda \eta}{hc} \left[\frac{4kT}{R_0 A}+2q^2 \eta \Phi_b\right]^{-1/2}$$

With q as the electronic charge, $$\lambda$$ 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, $$R_0A$$ 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 $$\Phi_b$$ 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 $$t_c$$.


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

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.


 * $$ \text{Signal}_{\text{avg}} = \frac{1}{N}\big( \sum_i^{N} \text{Signal}_i \big) $$


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

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.


 * $$R = \frac{\text{Signal}_{\text{avg}}}{H G} = \frac{\text{Signal}_{\text{avg}}}{\int dH dA_d d\Omega_{BB}}$$

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 area
 * $$G$$ is the total integrated etendue between the emitting source and detector surface
 * $$A_d$$ 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.


 * $$ \text{NEP} = \frac{\text{Noise}_{\text{rms}}}{R} = \frac{\text{Noise}_{\text{rms}}}{\text{Signal}_{\text{avg}}}H G $$

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.


 * $$ D^* = \frac{\sqrt{\Delta f A_d}}{\text{NEP}} = \frac{\sqrt{\Delta f A_d}}{H G} \frac{\text{Signal}_{\text{avg}}}{\text{Noise}_{\text{rms}}} $$