- Source: 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
∗
=
A
Δ
f
N
E
P
{\displaystyle D^{*}={\frac {\sqrt {A\Delta f}}{NEP}}}
, where
A
{\displaystyle A}
is the area of the photosensitive region of the detector,
Δ
f
{\displaystyle \Delta f}
is the bandwidth, and NEP the noise equivalent power in units [W]. It is commonly expressed in Jones units (
c
m
⋅
H
z
/
W
{\displaystyle cm\cdot {\sqrt {Hz}}/W}
) in honor of Robert Clark Jones who originally defined it.
Given that noise-equivalent power can be expressed as a function of the responsivity
R
{\displaystyle {\mathfrak {R}}}
(in units of
A
/
W
{\displaystyle A/W}
or
V
/
W
{\displaystyle V/W}
) and the noise spectral density
S
n
{\displaystyle S_{n}}
(in units of
A
/
H
z
1
/
2
{\displaystyle A/Hz^{1/2}}
or
V
/
H
z
1
/
2
{\displaystyle V/Hz^{1/2}}
) as
N
E
P
=
S
n
R
{\displaystyle NEP={\frac {S_{n}}{\mathfrak {R}}}}
, it is common to see the specific detectivity expressed as
D
∗
=
R
⋅
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.
D
∗
=
q
λ
η
h
c
[
4
k
T
R
0
A
+
2
q
2
η
Φ
b
]
−
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,
λ
{\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,
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),
η
{\displaystyle \eta }
is the quantum efficiency of the device, and
Φ
b
{\displaystyle \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
Δ
f
{\displaystyle \Delta f}
directly from the integration time constant
t
c
{\displaystyle t_{c}}
.
Δ
f
=
1
2
t
c
{\displaystyle \Delta f={\frac {1}{2t_{c}}}}
Next, an average signal and rms noise needs to be measured from a set of
N
{\displaystyle N}
frames. This is done either directly by the instrument, or done as post-processing.
Signal
avg
=
1
N
(
∑
i
N
Signal
i
)
{\displaystyle {\text{Signal}}_{\text{avg}}={\frac {1}{N}}{\big (}\sum _{i}^{N}{\text{Signal}}_{i}{\big )}}
Noise
rms
=
1
N
∑
i
N
(
Signal
i
−
Signal
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
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
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.
R
=
Signal
avg
H
G
=
Signal
avg
∫
d
H
d
A
d
d
Ω
B
B
,
{\displaystyle R={\frac {{\text{Signal}}_{\text{avg}}}{HG}}={\frac {{\text{Signal}}_{\text{avg}}}{\int dHdA_{d}d\Omega _{BB}}},}
where
R
{\displaystyle R}
is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
H
{\displaystyle H}
is the outgoing radiance from the black body (or light source) in W/sr/cm2 of emitting area
G
{\displaystyle G}
is the total integrated etendue between the emitting source and detector surface
A
d
{\displaystyle A_{d}}
is the detector area
Ω
B
B
{\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.
NEP
=
Noise
rms
R
=
Noise
rms
Signal
avg
H
G
{\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.
D
∗
=
Δ
f
A
d
NEP
=
Δ
f
A
d
H
G
Signal
avg
Noise
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}}}}}
See also
Sensitivity (electronics)
References
This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.
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