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
f
N
E
P
, where
A
is the area of the photosensitive region of the detector and
f
is the frequency bandwidth. It is commonly expressed in Jones units (
c
m
⋅
H
z
/
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
(in units of
A
/
W
or
V
/
W
) and the noise spectral density
S
n
(in units of
A
/
H
z
1
/
2
or
V
/
H
z
1
/
2
) as
N
E
P
=
S
n
R
, it's common to see the specific detectivity expressed as
D
∗
=
R
⋅
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
With q as the electronic charge,
λ
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,
R
0
A
is the zero-bias dynamic resistance area product (often measured experimentally, but also expressible in noise level assumptions),
η
is the quantum efficiency of the device, and
Φ
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 wavelet will be integrated over a given time constant over a given number of frames.
In detail, we compute the bandwidth
Δ
f
directly from the integration time constant
t
c
.
Δ
f
=
1
2
t
c
Next, an rms signal and noise needs to be measured from a set of
N
frames. This is done either directly by the instrument, or done as post-processing.
S
i
g
n
a
l
r
m
s
=
1
N
(
∑
i
N
S
i
g
n
a
l
i
2
)
N
o
i
s
e
r
m
s
=
σ
2
=
1
N
∑
i
N
(
S
i
g
n
a
l
i
−
S
i
g
n
a
l
a
v
g
)
2
Now, the computation of the radiance
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
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
=
S
i
g
n
a
l
r
m
s
H
G
=
S
i
g
n
a
l
∫
d
H
d
A
d
d
Ω
B
B
Where,
R
is the responsivity in units of Signal / W, (or sometimes V/W or A/W)
H
is the outgoing radiance from the black body (or light source) in W/sr/cm² of emitting area
G
is the total integrated etendue between the emitting source and detector surface
A
d
is the detector area
Ω
B
B
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.
N
E
P
=
N
o
i
s
e
r
m
s
R
=
N
o
i
s
e
r
m
s
S
i
g
n
a
l
r
m
s
H
G
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
N
E
P
=
Δ
f
A
d
H
G
S
i
g
n
a
l
r
m
s
N
o
i
s
e
r
m
s
