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 ${\displaystyle D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}}${\displaystyle D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}}, where ${\displaystyle A}${\displaystyle A} is the area of the photosensitive region of the detector, ${\displaystyle \Delta f}${\displaystyle \Delta f} is the bandwidth, and NEP the noise equivalent power [unit: ${\displaystyle W/{\sqrt {Hz}}}${\displaystyle W/{\sqrt {Hz}}}]. It is commonly expressed in Jones units (${\displaystyle cm\cdot {\sqrt {Hz}}/W}${\displaystyle cm\cdot {\sqrt {Hz}}/W}) in honor of Robert Clark Jones who originally defined it.^{[1] [2]}

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

${\displaystyle D^{*}={\frac {q\lambda \eta }{hc}}\left[{\frac {4kT}{R_{0}A}}+2q^{2}\eta \Phi _{b}\right]^{-1/2}}${\displaystyle 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, ${\displaystyle \lambda }${\displaystyle \lambda } is the wavelength of interest, h is the Planck constant, c is the speed of light, k is the Boltzmann constant, T is the temperature of the detector, ${\displaystyle R_{0}A}${\displaystyle R_{0}A} is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions), ${\displaystyle \eta }${\displaystyle \eta } is the quantum efficiency of the device, and ${\displaystyle \Phi _{b}}${\displaystyle \Phi _{b}} is the total flux of the source (often a blackbody) in photons/sec/cm².

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 ${\displaystyle \Delta f}${\displaystyle \Delta f} directly from the integration time constant ${\displaystyle t_{c}}${\displaystyle t_{c}}.

${\displaystyle \Delta f={\frac {1}{2t_{c}}}}${\displaystyle \Delta f={\frac {1}{2t_{c}}}}

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

${\displaystyle {\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big (}\sum _{i}^{N}{\text{Signal}}_{i}{\big )}}${\displaystyle {\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big (}\sum _{i}^{N}{\text{Signal}}_{i}{\big )}}

${\displaystyle {\text{Noise}}_{\text{rms}}={\sqrt {{\frac {1}{N}}\sum _{i}^{N}({\text{Signal}}_{i}-{\text{Signal}}_{\text{avg}})^{2}}}}${\displaystyle {\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 ${\displaystyle H}${\displaystyle 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 ${\displaystyle A_{d}}${\displaystyle 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.

${\displaystyle R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}},}${\displaystyle R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}},}

where

• ${\displaystyle R}${\displaystyle R} is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
• ${\displaystyle H}${\displaystyle H} is the outgoing radiance from the black body (or light source) in W/sr/cm² of emitting area
• ${\displaystyle G}${\displaystyle G} is the total integrated etendue between the emitting source and detector surface
• ${\displaystyle A_{d}}${\displaystyle A_{d}} is the detector area
• ${\displaystyle \Omega _{BB}}${\displaystyle \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.

${\displaystyle {\text{NEP}}={\frac {{\text{Noise}}_{\text{rms}}}{R}}={\frac {{\text{Noise}}_{\text{rms}}}{{\text{Signal}}_{\text{avg}}}}HG}${\displaystyle {\text{NEP}}={\frac {{\text{Noise}}_{\text{rms}}}{R}}={\frac {{\text{Noise}}_{\text{rms}}}{{\text{Signal}}_{\text{avg}}}}HG}

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.

${\displaystyle D^{*}={\frac {\sqrt {\Delta fA_{d}}}{\text{NEP}}}={\frac {\sqrt {\Delta fA_{d}}}{HG}}{\frac {{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}}}${\displaystyle D^{*}={\frac {\sqrt {\Delta fA_{d}}}{\text{NEP}}}={\frac {\sqrt {\Delta fA_{d}}}{HG}}{\frac {{\text{Signal}}_{\text{avg}}}{{\text{Noise}}_{\text{rms}}}}}

• Sensitivity (electronics)

Public Domain This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022年01月22日.

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