Specific detectivity (D*)

Specific detectivity, written D* and read "D-star", measures how weak a signal a detector can sense, normalized so that detectors of different size can be compared. It is the inverse of the noise equivalent power, scaled by the square root of the detector area $A$ and the measurement bandwidth $\Delta f$:

$$D^* = \frac{\sqrt{A\,\Delta f}}{\mathrm{NEP}}$$

With $A$ in cm² and NEP in watts for bandwidth $\Delta f$ (or in W/√Hz with $\Delta f$ = 1 Hz), D* comes out in cm·Hz^½/W, a unit named the Jones after R. Clark Jones, who introduced D*. Higher is better. A 1 mm × 1 mm detector ($A$ = 0.01 cm²) with an NEP of 1 pW/√Hz has D* = 0.1/10⁻¹² = 10¹¹ Jones. Orders of magnitude for common detector families: thermopiles and pyroelectric detectors around 10⁸–10⁹ Jones, cooled HgCdTe photodiodes around 10¹⁰–10¹¹ in the long-wave infrared, and silicon and InGaAs photodiodes 10¹² or more near their peak response.

Why normalize by area

For many infrared detectors the dominant noise sources, generation-recombination in the bulk, diffusion current, and fluctuations in absorbed background photons, all scale with area, so their noise current grows as $\sqrt{A}$ and so does their NEP. Dividing it out gives a property of the material and design rather than of the chip size, and it lets a 50 µm pixel and a 5 mm single element be compared on one scale. The $\sqrt{\Delta f}$ factor does the same for bandwidth, since white noise power is proportional to it.

The normalization is only valid when those assumptions hold. When the noise is set by a transimpedance amplifier, by the load resistor's thermal noise, or by 1/f noise, NEP does not scale as $\sqrt{A}$, and D* for a small detector overstates what a larger one of the same type would do. For visible and near-infrared photodiodes, where amplifier noise usually dominates, NEP is the more useful number.

How it is quoted and measured

D* depends on wavelength, modulation frequency and bandwidth, and full specifications give all three: D*(λ, f, Δf), for example D*(10 µm, 1 kHz, 1 Hz). Older infrared literature also quotes a blackbody D*, D*(500 K, f, Δf), measured with a 500 K blackbody source and therefore, for a photon detector, lower than the peak spectral value. The measurement follows the NEP procedure described in photodetector characterization: a calibrated, chopped source of known power, a lock-in amplifier or spectrum analyzer to measure signal and noise in a known bandwidth, and a known active area. The field of view and the background temperature must be stated, since background photon noise often dominates.

Background limit

A photon detector looking at a 300 K scene through a hemispherical field of view is ultimately limited by the random arrival of background photons, the BLIP condition. For an ideal photovoltaic detector of quantum efficiency $\eta$,

$$D^*_{\mathrm{BLIP}} = \frac{\lambda}{hc}\sqrt{\frac{\eta}{2Q_B}},$$

where $Q_B$ is the background photon flux per unit area within the detector's spectral band (the values below take $\eta$ = 1). With a 10 µm cutoff, $Q_B \approx 4.9 \times 10^{17}$ photons/(cm²·s) and D*_BLIP ≈ 5.1 × 10¹⁰ Jones; with a 5 µm cutoff the background is weaker and the limit rises to about 1.5 × 10¹¹. A photoconductor, which adds recombination noise, is lower by $\sqrt{2}$. Cold shields that narrow the field of view raise the limit, which is why infrared focal plane arrays sit behind a cooled aperture.

Thermal detectors are limited by temperature fluctuations from radiative exchange with the surroundings. For a black detector at 300 K viewing a 300 K background over a hemisphere, the limit is

$$D^* = \left[8k\sigma\,(T_d^5 + T_b^5)\right]^{-1/2}$$ $$\approx 1.8 \times 10^{10}\ \mathrm{Jones}.$$

Practical thermopiles and bolometers at room temperature typically sit one to two orders of magnitude below it.

Common questions

What is a Jones unit?

One Jones is 1 cm·Hz^½/W, the unit of D*. It is used only for detectivity.

How is D* converted to NEP?

NEP = √A / D* in W/√Hz, with $A$ in cm². A detector of 0.25 mm² ($A$ = 0.0025 cm²) with D* = 10¹⁰ Jones has an NEP of 5 pW/√Hz.

Is higher D* always better?

For the same area and bandwidth, yes. Across sizes, D* hides the fact that a larger detector has a higher NEP; a system that needs a large collecting area may prefer a lower-D* detector that is available in that size, and a fast system cares about bandwidth, which D* removes.

References: E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, 1996); A. Rogalski, Infrared and Terahertz Detectors, 3rd ed. (CRC Press, 2019); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).