Modern detection systems are increasingly limited in sensitivity by the background thermal photons which enter the receiving system. Expressions for the fluctuations of detected thermal radiation are derived. Incoherent and heterodyne detection processes are considered. References to the subject of photon detection statistics are given

Gulkis, S.

English

NASA Technical Reports Server

14330
TDA Progress Report 42-71 July - September 1982
Thermal Background Noise Limitations
S. Gulkis
Atmospheric Sciences Section
Modern detection systems are increasingly limited in sensitivity by the background
thermal photons which enter the receiving system. This paper derives expressions for the
fluctuations of detected thermal radiation. Incoherent and heterodyne detection pro-
cesses are considered. The paper is intended to be tutorial in style. Many good references
to the subject of photon detection statistics are given.
I. Introduction
An important consideration in evaluating and designing
sensitive receiving systems for astronomy, deep space commu-
nications, and other related activities (such as the search for
extraterrestrial intelligence), in which the goal is to detect the
presence of a weak signal in the presence of noise, is the
evaluation of the noise. In general, noise is due to the com-
bined presence of both the signal and various sources of
unwanted noise. The unwanted noise may arise in the detector
itself, from the immediate surroundings of the detector (i.e.,
emission from the telescope), or from the (background) emis-
sion which enters the telescope and is intercepted by the
detector. Background emission may originate from the atmos-
phere, from nearby objects, or from the cosmic background
itself.
When the detection device is a square-law detector, the
noise manifests itself as fluctuations in the output voltage (or
current) from the detector V(t}, where
In this expression P(t) is the instantaneous power incident on
the detector, and r is a constant of proportionality.
A measure of the fluctuations in V(t) about its mean is the
variance defined by
var[F(r)J = i r lim - I
r-x» J -
= r2 lim ±,
+T/2
772
V(t) = rP(t) (1)
(P(t}-P?dt (2)
This parameter is frequently encountered in estimating the
signal-to-noise ratio of a system. In general, a calculation of
the variance requires a knowledge of both the signal and noise,
as well as the frequency response of the system.
Frequently, there is a desire to know the variance when/If/)
is due to a finite band of thermal radiation at temperature T.
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In particular, this result is useful when the dominating source
of noise is bandlimited thermal radiation whose power greatly
exceeds the signal power. This situation is commonly referred
to as the background limited noise condition. Numerous
approximations of the background limited noise appear in the
literature, especially in the thermal limit, where hvjkT « 1
and in the quantum limit where hv/kT » 1. However, the
more general expressions which describe thermal radiation
over the entire range of hvjkT are not as readily available. In
this note, I derive some useful expressions related to the
detection of thermal radiation and discuss some of the results.
The purpose of the paper was primarily to acquaint the author
with the trends of background limited noise. It is printed here
in the anticipation that others may find the discussion useful.
Many important points related to the detection process and
signal/noise ratios are not covered in this note. In particular, I
omitted all discussions of the type of detector involved in the
detector process. The reader is referred to the Bibliography for
additional discussions.
II. Derivations
A. Photons as Independent Particles
We assume initially that the radiation field incident on the
detector is a stream of statistically independent photons. For
this situation, each photon produces a power
p(t) = h i > S ( t - t n ) (3)
and an energy given by
E = p(t)dt = hv (4)
Equation (3) is not strictly correct since photons are bosons
and consequently do not occupy available energy states inde-
pendently. In particular, the statistics of arrival times of
photons at a detector indicate that photons are "bunched".
Nevertheless, we will continue with this assumption and apply
a correction for the bunching later on. The justification for
this approach is threefold: ( l ) i t is relatively straightforward;
(2) it introduces the terms in the results in a clear manner; and
(3) the analysis leads to the quantum or particle-like noise
term.
Since the total power received by the detector is a linear
superposition of the sum of all the photons reaching the
detector during the interval (0,0), we can write for the total
power received by the detector when K photons arrive
PK = (5)
where t^ is the arrival time of the k photon. These arrival times
are random and unknown.
Series such as that given by Eq. (5) can be evaluated using
Campbell's theorem, which is discussed in detail by Rice
(Ref. 1). Campbell's theorem states that (1) the average value
of P(t) (averaged over both time and over all values of k) is
given by
P(t)
••£ (6)
and (2) the variance is given by
var [P(t)] = AP2 = P*(t)dt (7)
where R is the average number of photon arrivals per second.
The principal quantity of interest in this note is the variance,
for this quantity is related to the precision of physical mea-
surements.
To evaluate the variance, we need to evaluate both R a'nd
the integral
P\t)dt
R can be calculated directly from Eq. (6), using Eq. (4) and
Planck's radiation law to calculate the average power. Planck's
law gives for the radiance of a blackbody radiator at tempera-
ture rand frequency c
B = 1 (8)
where
B = radiance, W m~2 Hz"1 rad~2
h = Planck's constant (= 6.63 X exp (-34) joule sec)
v = frequency, Hz
c = velocity of light, (= 3 X exp (108) m/sec"1)
k = Boltzmann's constant, (= 1.38 X exp (10~23) joule
T = temperature, K
54
Thus a blackbody source of emission at temperature T pro-
duces a power given by
P = ASlBdv (9)
where £2 is the solid angle subtended by the source, A is the
effective collecting area of the detector (telescope), and dv is
the bandwidth of the received radiation. Thus we have for R
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OF-POOR QUAUTY
B. Photons as Bosons
We now consider the problem of the nonindependence of
the photons. In a Bose-Einstein system composed of weakly
interacting photons at thermal equilibrium, the average num-
ber of photons in a given energy state (the occupancy) is given
by (Ref. 2)
*=^ = 2W±)2hv \cl
- 1
(10)
To evaluate the integral in Eq. (7), we use Parseval's theo-
rem to rewrite it as
A/>2 = 2R \S(f)\2 df (11)
where 5 (/) is the Fourier transform of p (t). Hence we have
\S(f)\ = I p(t) e~2niftdt = hv (12)
J — 00
The variance or mean square value of the power fluctuation is
derived by combining Eqs. (11) and (12) to yield
(13)AP2 = 2R(hv)2 I df
If there is a postdetection filter in the system which passes
a range of frequencies A/, then the power content of the
fluctuating power is given by
AP2 = 2R(hvY A/
Substituting for/? in Eq. (14), we obtain
(14)
AP2 = (7)' 1ef">/kT- (hv)2 dv&f (15)
It is of interest to note that the familiar shot noise equation
can be derived from Eq. (14) by substituting E = h v for the
photon energy and P = Rhv for the average power delivered to
the detector. This result is a consequence of the fact that shot
noise is related to the statistics of independent particles (our
initial assumption), which obey Poisson statistics. Thus we
have
= (2EPA/)1/2 (16)
n =
1
e""/kT-
and the mean square fluctuations are given by
= n ( n + 1)
(17)
(18)
An analysis which takes into account the Bose-Einstein statis-
tics leads to a variance given by (Refs. 3, 4)
AP2 = (19)
Comparing this expression with Eq. (15) reveals that the
fluctuations are higher by (1 + n). The independent photon
calculations are corrected for Bose-Einstein statistics by multi-
plying by the factor (n + 1). The term (n + 1) is known as the
Bose factor. This factor is a monotonically decreasing function
of hv/KT, approaching unity for large values of Hv/kT.
Photons behave like individual particles in this spectral region.
In the Rayleigh- Jeans limit (kT»hv), the fluctuations are
larger than those of independent particles. This increased noise
power is due to the wavelike nature of photons. It results from
each mode in the field beating with itself to produce a mean
square power fluctuation. In the Rayleigh-Jeans limit, the
variance becomes
(20)
(21)
A/>2 = 4J14 l (kT)2 dvAf
We obtain the classical low-frequency approximation
AP =
by taking
and by defining P = kTdv and A/= l/2t.
C. Noise Equivalent Power
A standard measure of sensitivity, used in the optical and
infrared spectral regions, is the noise equivalent power (NEP),
55
ORIGINAL PAGE Si
; OF,.POOR QUALITY
defined as the incident signal power required to produce a
detector signal equal to the rms noise power in a 1-Hz post-
detection bandwidth. We have from Eq. (19)
NEP = 4VA (-} O)2 n (n + 1) di>
W/
1/2.
(22)
The units of NEP are watts/\/Hz, although occasionally
authors incorrectly drop the \/Hz term. The reference band-
width dv, field of view SI, and detector area A should be
specified with each NEP.
The NEP reduces to the following expression in the thermal
limit
NEP = 2kT /S14 - dv (23)
D. Heterodyne Detection
We consider here a square law mixer as defined by Eq. (1).
For a heterodyne receiver, P(t) is given by (e.g., Blaney,
Ref. 5)
only to the random arrival times of the photons. Poisson
statistics apply since the phase of the local oscillator power is
fixed. The second point is that the local oscillator power is a
multiplicative factor in the first two terms but not the third.
Since the "signal" in a heterodyne system is multiplied by the
local oscillator power, the local oscillator power may be
increased to the point where the background power term is
negligible. For this condition we have the following
<*P L hvdv+P L hvASl(^ \ ndv (27)
III. Discussion
The principal results derived above for the rms noise power
which results from the detection of blackbody radiation are as
follows:
Incoherent detection (rms power)
AP = (3 *>* (28)
P(t) = [(2PL)1'2 cos uLt + (2PB)'/2 cos (uat - 0)]2 (24) Heterodyne detection (spectral power)
where PL is the local oscillator power (single polarization), PB
is the signal power, and <j> is the phase difference between the
local oscillator and the signal. As in our previous discussion,
we assume that PB arises from the radiation from a thermal
source inside the antenna pattern. Hence we are interested in
the noise fluctuations due to coherently detected thermal
radiation.
The resultant form of P(t) on dropping high-frequency
terms is
(25)
The first two terms create a dc current in the mixer while the
last term produces the IF current. The variance of P(t)
includes the sum of the variances of the local oscillator power,
the background power, and the cross product of the local
oscillator voltage and the background voltage. The resultant
noise power has the form
\ /"\2-l1 +ASI {-} n \ d v
L W/ J
(29)
These equations are frequently interpreted in terms of
either the NEP or in terms of an equivalent temperature. The
equivalent temperature representation "of the incoherent detec-
tion process follows from taking the input and output band-
widths to be equal and the beam filling factor to be unity. It
follows that Eqs. (28) and (29) can then be written as
AP = n(n + \ )dv
n+ l)dv
(30)
(31)
Noting that these equations are now in the form of a spectral
density times a bandwidth, the equivalent temperature is
defined as follows:
(32)
ndv + (NEP)2 A/
(26) TE =T( (33)
Two important points to note about this equation are the
following. The variance due to the local oscillator power is due
Figures 1 and 2 show equivalent temperatures TE for inco-
herent and heterodyne detectors. We note from these figures
56
that in the thermal limit the equivalent temperature equals the noise power following a heterodyne detector if the frequency
source temperature for both the incoherent and linear ampli- is sufficiently high and the background emission low. In the
fier cases. However, in the quantum limit, the linear amplifier radio spectral region, the 2.7 K cosmic background radiation
noise increases like hv, whereas the incoherent noise power provides a natural noise source and there is no disadvantage of
decreases. Thus, incoherent noise power is much less than the using a heterodyne detector (or a linear amplifier).
Acknowledgments
I would like to thank R. Beer, R. W. Boyd, C. Stelzried, P. Swanson, and R. Menzies
for their very helpful comments and for leading me to a number of useful references.
References
1. Rice, S. 0., "Mathematical Analysis of Random Noise," Bell System Tech. J., Vol. 23,
July 1944, Vol. 24, Jan. 1945.
2. Tolman, R. C., The Principles of Statistical Mechanics, Oxford University Press, 1938.
3. Boyd, R. W., "Photon Bunching and the Photon-Noise-Limited Performance of Infra-
red Detectors," InfraredPhys., Vol. 22, pp. 157- 162, 1982.
4. Kingston, R. H., Detection of Optical and Infrared Radiation, Springer-Verlag, 1978.
5. Blaney, T. G., "Signal-To-Noise Ratio and Other Characteristics of Heterodyne Radia-
tion Receivers," Space Science Reviews, Vol. 17, pp. 691-701, 1975.
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58
ORIGINAL PAGE IS
OF POOR QUALITY
'*.•*
' 4
1 2 3 4 5
LOG(F), GHz
Fig. 1. Equivalent temperatures of incoherent detected thermal noise
for a number of different background temperatures for 2 to 512 K
o
o
2 3
LOG(F), GHz
Fig. 2. Equivalent temperatures of heterodyne detected thermal noise
for a number of different background temperatures from 2 to 512 K
59


Thermal background noise limitations

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