These three quantities are
of importance in optical sensing systems and concern the measurement of
radiation intensity. NEP and D* may
be applied to the whole spectrum of electromagnetic radiation; NETD is only
relevant in the thermal infrared region.
NEP, D*, NETD are applied to both image sensors and single
detectors. When applied to an image
sensors, one considers a single pixel from the array.
Definitions in words
|
NEP |
noise equivalent power: the signal [V] of that optical power [W] impinging on a pixel that corresponds to the RMS value of the noise [VRMS]. |
|
D* |
Specific detectivity; noise equivalent power, normalized to unity detector (pixel) area (to compare pixels of different sizes) and to unity bandwidth (to compare systems of different speeds). |
|
NETD |
noise equivalent temperature difference: the signal [V] of that scene temperature difference [K] impinging on a pixel that corresponds to the RMS value of the noise [VRMS]. This is of course only relevant for thermal IR detectors. |
As you are an engineer, you
want something more solid than a definition in words. Herebelow
follow the formulas.
Let
R = responsivity of detector channel (e.g. a pixel) in [V/W]
Z = optics conversion factor in [K/W]
A = pixel area [m2]
Vn = RMS noise voltage in [V]
BW = signal bandwidth, as seen by the final output, in [Hz]
Fa = acquisition frame rate in [Hz]
(clearly distinguished from BW!)
SVn = spectral density of noise voltage in [V2/Hz]
In general Vn is the integral over
bandwidth of (SVn(f).df). For white noise: (Vn)2 = SVn * BW.
then:
NEP = Vn / R in [W]
NETD = NEP * Z in [K]
According to this
definition, NEP and NETD are not normalized quantities; their meaning is
"the accuracy of a single measurement of a pixel of such-and-such
camera". Some authors consider NEP and NETD as normalized
. To avoid confusion, we will postfix the normalized qualtities
with a "*" (as with D*) [1]
NEP* = NEP / sqrt(Fa) in [W/sqrt(Hz)] [2]
NETD* = NETD / sqrt(Fa) in [K/sqrt(Hz)]
D* = sqrt(A)/NEP* in [m.sqrt(Hz)/W]
A chopping
image sensor takes alternating illuminated and dark (reference) frames. The
resulting "real" image is obtained from the difference (demodulation)
of two consecutive frames. Chopping removes many sources of non-uniformity and
low frequency (LF) noise as 1/f noise. As the resulting images (pixel signals)
are uncorrelated in time, and have thus white noise, it is again possible to
calculate NEP*, NETD* and D*.
Chopping and demodulation remind of correlated double sampling.
The chopping and
differencing operations decrease the signal to noise ratio [for a white noise
limited system] with a factor 2, compared to a situation where 100% of the time
is used for useful observation
consider the following
case:
Note: Fa
is the well defined image acquisition frequency, in
this case 40Hz. BWa is the signal bandwidth as
supposed for a, which may be much higher, e.g. in case of a bolometer readout,
it is the bandwidth of the pixel sampling. Clever readers will have seen
different definitions for NEP* and D*, featuring a bandwidth and not a sampling
frequency. Image sensors and pixels are sampled data systems. In cases that Fa and BW do strongly differ, a
image sensor is indeed inferior to a continuous time system. Think it over.
Note: chopping decreases
the S/N by a factor 2 if the signal is white noise limited. even if the unchopped signal contains 1/f or other LF noise, it will
produce a signal that is free of 1/f noise, and thus can be averaged to improve
S/N. Although the chopped signal is free of LF noise, NEPb
as given above does retain its 1/f contribution! The
1/f is just aliased to higher frequencies. It would be completely wrong to
assume that 1/f noise sources may be neglected when using chopping.
Note: capacitor
(dis-)charge noise (also called reset noise or kTC-noise)
may be cancelled by chopping, if and only if the circuit operation is such that
the capacitor is not electrically (dis-)charged in between the chopper halve
periods. This principle is similar to correlated double sampling.
A 300K
blackbody emits about P = 1 kW/m2. This amount follows a 4th law, thus P ~ T4.
The differential power per kelvin is thus
dP 4 1kW/m2
-- = ---.------ = 13 W/m2KdT 300 K
Caveat: here we assumed the full
optical bandwidth, and a perfect blackbody. A real object has a lower emission
(typ 50%), does not follow the 4th power law
precisely, and the spectrum is typically limited by the detector response,
filter transmission or atmospheric transmission. In literature a value of 2.5 W/m2K
for the 8-12 um band seems to be popular.
The object emits power in all
directions, i.e. over a halve sphere. At a distance r of the source, the power
density is 2.PI.r2. The pixel in the camera sees the outside through the
opening of the lens. It is of no importance whether the lens is in place or
just an opening, as the average power through the opening is the same. With a
lens diameter d and a focal distance F, one defines the "f-number" of
the lens as f = F/d. "Good" (sensitive) lenses have low f. The
highest qualities are in the range 0.8...1.5. The lens only transmits a part of
the total radiation of the hemisphere. In the case of uniform, omnidirectional
radiation, the ratio between the energy passgin
through the lens opening and the total radiation density is 1/(4.f2). The
attenuation of the radiation flux from object radiation flux to detector
incoming power is thus:
power_pixel[W] = flux_scene[W/m2]
. 1/(4.f^2) Area_detector [W/K] [W/m2K]
EXAMPLE suppose 300K blackbody
scene, f=1 optics, 20x20 um detector. For the 300K scene we calculate 13 W/m2K
differential power. thus power_pixel[W/K] = 13
[W/m2K] . (1/4) . 400E-12[m2] = 1 [nW/K]. Btw, this
example shows that any reasonably good NEP should be better than 1E-10W.