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 <math>D^*=\frac{\sqrt{A \Delta f}}{NEP}</math>, where <math>A</math> is the area of the photosensitive region of the detector, <math>\Delta f</math> is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units (<math>cm \cdot \sqrt{Hz}/ W</math>) in honor of Robert Clark Jones who originally defined it.<ref>R. C. Jones, "Quantum efficiency of photoconductors," Proc. IRIS 2, 9 (1957)</ref><ref>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)</ref>

Given that noise-equivalent power can be expressed as a function of the responsivity <math>\mathfrak{R}</math> (in units of <math>A/W</math> or <math>V/W</math>) and the noise spectral density <math>S_n</math> (in units of <math>A/Hz^{1/2}</math> or <math>V/Hz^{1/2}</math>) as <math>NEP=\frac{S_n}{\mathfrak{R}}</math>, it is common to see the specific detectivity expressed as <math>D^*=\frac{\mathfrak{R}\cdot\sqrt{A}}{S_n}</math>.

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.

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

With q as the electronic charge, <math>\lambda</math> 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, <math>R_0A</math> is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), <math>\eta</math> is the quantum efficiency of the device, and <math>\Phi_b</math> 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 <math>\Delta f</math> directly from the integration time constant <math>t_c</math>.

<math> \Delta f = \frac{1}{2 t_c} </math>

Next, an average signal and rms noise needs to be measured from a set of <math>N</math> frames. This is done either directly by the instrument, or done as post-processing.

<math> \text{Signal}_{\text{avg}} = \frac{1}{N}\big( \sum_i^{N} \text{Signal}_i \big) </math>

<math> \text{Noise}_{\text{rms}} = \sqrt{\frac{1}{N}\sum_i^N (\text{Signal}_i - \text{Signal}_{\text{avg}})^2} </math>

Now, the computation of the radiance <math>H</math> 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 <math>A_d</math> 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.

<math>R = \frac{\text{Signal}_{\text{avg}}}{H G} = \frac{\text{Signal}_{\text{avg}}}{\int dH dA_d d\Omega_{BB}}</math>

Where,

• <math>R</math> is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
• <math>H</math> is the outgoing radiance from the black body (or light source) in W/sr/cm² of emitting area
• <math>G</math> is the total integrated etendue between the emitting source and detector surface
• <math>A_d</math> is the detector area
• <math>\Omega_{BB}</math> 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.

<math> \text{NEP} = \frac{\text{Noise}_{\text{rms}}}{R} = \frac{\text{Noise}_{\text{rms}}}{\text{Signal}_{\text{avg}}}H G </math>

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.

<math> 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}}} </math>

References

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.