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 $D^*=\frac$, 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 It is commonly expressed in ''Jones'' units ($cm \cdot \sqrt/ W$) in honor of Robert Clark Jones who originally defined it.R. C. Jones, "Proposal of the detectivity D** for detectors limited by radiation noise," ''J. Opt. Soc. Am.'' 50, 1058 (1960), )

Given that noise-equivalent power can be expressed as a function of the responsivity $\mathfrak$ (in units of $A/W$ or $V/W$) and the noise spectral density $S_n$ (in units of $A/Hz^$ or $V/Hz^$) as $NEP=\frac$, it is common to see the specific detectivity expressed as $D^*=\frac$.

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 \left frac+2q^2 \eta \Phi_b\right

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 $t_c$.

:$\Delta f = \frac$

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.

:$\text_ = \frac\big( \sum_i^ \text_i \big)$

:$\text_ = \sqrt$

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.

:$R = \frac = \frac$

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
* $\Omega_$ 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.

:$\text = \frac = \fracH G$

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.

:$D^* = \frac = \frac \frac$

References

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Physical quantities
Infrared imaging