Physics:Specific detectivity

Short description: Parameter characterizing photodetector performance

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]\displaystyle{ D^*=\frac{\sqrt{A \Delta f}}{NEP} }[/math], where [math]\displaystyle{ A }[/math] is the area of the photosensitive region of the detector, [math]\displaystyle{ \Delta f }[/math] is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units ([math]\displaystyle{ cm \cdot \sqrt{Hz}/ W }[/math]) in honor of Robert Clark Jones who originally defined it.

Given that noise-equivalent power can be expressed as a function of the responsivity [math]\displaystyle{ \mathfrak{R} }[/math] (in units of [math]\displaystyle{ A/W }[/math] or [math]\displaystyle{ V/W }[/math]) and the noise spectral density [math]\displaystyle{ S_n }[/math] (in units of [math]\displaystyle{ A/Hz^{1/2} }[/math] or [math]\displaystyle{ V/Hz^{1/2} }[/math]) as [math]\displaystyle{ NEP=\frac{S_n}{\mathfrak{R}} }[/math], it is common to see the specific detectivity expressed as [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ R_0A }[/math] is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), [math]\displaystyle{ \eta }[/math] is the quantum efficiency of the device, and [math]\displaystyle{ \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]\displaystyle{ \Delta f }[/math] directly from the integration time constant [math]\displaystyle{ t_c }[/math].

[math]\displaystyle{ \Delta f = \frac{1}{2 t_c} }[/math]

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

[math]\displaystyle{ \text{Signal}_{\text{avg}} = \frac{1}{N}\big( \sum_i^{N} \text{Signal}_i \big) }[/math]

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

Where,

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