Shot noise or Poisson noise is a type of electronic noise which can be modeled by
a Poisson process. In electronics shot noise originates from the discrete
nature of electric charge. Shot noise also occurs in photon counting in
optical devices, where shot noise is associated with the particle nature of
light. Origin
It is known that in a statistical experiment such as tossing a fair coin
and counting the occurrences of heads and tails, the numbers of heads and
tails after a great many throws will differ by only a tiny percentage, while
after only a few throws outcomes with a significant excess of heads over tails
or vice versa are common; if an experiment with a few throws is repeated
over and over, the outcomes will fluctuate a lot.
Shot noise exists because phenomena such as light and electric current consist of
the movement of discrete 'packets'. Consider light—a stream of discrete
photons—coming out of a laser pointer and hitting a wall to create a visible
spot. The fundamental physical processes that govern light emission are such that
these photons are emitted from the laser at random times; but the many billions
of photons needed to create a spot are so many that the brightness, the number
of photons per unit time, varies only infinitesimally with time. However, if
the laser brightness is reduced until only a handful of photons hit the wall
every second, the relative fluctuations in number of photons, i.e., brightness,
will be significant, just as when tossing a coin a few times. These
fluctuations are shot noise. The concept of shot noise was first
introduced in 1918 by Walter Schottky who studied fluctuations of current in
vacuum tubes. Shot noise may be dominant when the
finite number of particles that carry energy is sufficiently small so that
uncertainties due to the Poisson distribution, which describes the
occurrence of independent random events, are of significance. It is important in
electronics, telecommunications, optical detection, and fundamental physics.
The term can also be used to describe any noise source, even if solely
mathematical, of similar origin. For instance, particle simulations may
produce a certain amount of "noise", where due to the small number of
particles simulated, the simulation exhibits undue statistical fluctuations
which don't reflect the real-world system. The magnitude of shot noise
increases according to the square root of the expected number of events, such
as the electric current or intensity of light. But since the strength of the
signal itself increases more rapidly, the relative proportion of shot noise
decreases and the signal to noise ratio increases anyway. Thus shot noise is
most frequently observed with small currents or low light intensities that
have been amplified. For large numbers, the Poisson
distribution approaches a normal distribution about its mean, and the
elementary events are no longer individually observed, typically making
shot noise in actual observations indistinguishable from true Gaussian
noise. Since the standard deviation of shot noise is equal to the square root
of the average number of events N, the signal-to-noise ratio is given by:
Thus when N is very large, the signal-to-noise ratio is very large as
well, and any relative fluctuations in N due to other sources are more likely to
dominate over shot noise. However when the other noise source is at a fixed
level, such as thermal noise, or grows slower than , increasing N can lead to
dominance of shot noise. Properties
= Electronic devices= Shot noise in electronic circuits
consists of random fluctuations of the electric current in a DC current which
originate due to fact that current actually consists of a flow of discrete
charges. Because the electron has such a tiny charge, however, shot noise is of
relative insignificance in many cases of electrical conduction. For instance 1
ampere of current consists of about 7018624000000000000♠6.24×1018 electrons
per second; even though this number will randomly vary by several billion in any
given second, such a fluctuation is minuscule compared to the current
itself. In addition, shot noise is often less significant as compared with two
other noise sources in electronic circuits, flicker noise and
Johnson–Nyquist noise. However, shot noise is temperature and frequency
independent, in contrast to Johnson–Nyquist noise, which is
proportional to temperature, and flicker noise, with the spectral density
decreasing with the frequency. Therefore at high frequencies and low temperatures
shot noise may become the dominant source of noise.
With very small currents and considering shorter time scales shot noise can be
significant. For instance, a microwave circuit operates on time scales of less
than a nanosecond and if we were to have a current of 16 nanoamperes that would
amount to only 100 electrons passing every nanosecond. According to Poisson
statistics the actual number of electrons in any nanosecond would vary
by 10 electrons rms, so that one sixth of the time less than 90 electrons would
pass a point and one sixth of the time more than 110 electrons would be counted
in a nanosecond. Now with this small current viewed on this time scale, the
shot noise amounts to 1/10 of the DC current itself.
The result by Schottky, based on the assumption that the statistics of
electrons passage is Poissonian, reads for the spectral noise density at the
frequency , where is the electron charge, and is
the average current created by the electron stream. The noise spectral
power is frequency independent, which means the noise is white. This is the
classical result in the sense that it does not take into account that
electrons obey Fermi–Dirac statistics. This can be combined with the Landauer
formula, which relates the average current with the transmission
eigenvalues of the contact through which the current is measured. In the
simplest case these transmission eigenvalues can be taken energy
independent, the Landauer formula is where is the applied voltage. This
provides for commonly referred to as the Poisson
value of shot noise, . The correct result takes into account the quantum
statistics of electrons and reads It was obtained in the 1990s by Khlus,
Lesovik, and Büttiker. This noise is white and is always suppressed with
respect to the Poisson value. The degree of suppression, , is known as the Fano
factor. Noises produced by different transport channels are independent.
Fully open and fully closed channels produce no noise, since there are no
irregularities in the electron stream. At finite temperature, a closed
expression for noise can be written as well. It interpolates between shot noise
and Nyquist-Johnson noise. Examples
Tunnel junction is characterized by low transmission in all transport channels,
therefore the electron flow is Poissonian, and the Fano factor equals
one. Quantum point contact is characterized
by an ideal transmission in all open channels, therefore it does not produce
any noise, and the Fano factor equals zero. The exception is the step between
plateaus, when one of the channels is partially open and produces noise.
A metallic diffusive wire has a Fano factor of 1/3 regardless of the geometry
and the details of the material. In 2DEG exhibiting fractional quantum
Hall effect electric current is carried by quasiparticles moving at the sample
edge whose charge is a rational fraction of the electron charge. The first direct
measurement of their charge was through the shot noise in the current.
Effects of interactions While this is the result when the
electrons contributing to the current occur completely randomly, unaffected by
each other, there are important cases in which these natural fluctuations are
largely suppressed due to a charge build up. Take the previous example in which
an average of 100 electrons go from point A to point B every nanosecond.
During the first half of a nanosecond we would expect 50 electrons to arrive at
point B on the average, but in a particular half nanosecond there might
well be 60 electrons which arrive there. This will create a more negative
electric charge at point B than average, and that extra charge will tend to repel
the further flow of electrons from leaving point A during the remaining
half nanosecond. Thus the net current integrated over a nanosecond will tend
more to stay near its average value of 100 electrons rather than exhibiting the
expected fluctuations we calculated. This is the case in ordinary metallic
wires and in metal film resistors, where shot noise is almost completely
cancelled due to this anti-correlation between the motion of individual
electrons, acting on each other through the coulomb force.
However this reduction in shot noise does not apply when the current results
from random events at a potential barrier which all the electrons must
overcome due to a random excitation, such as by thermal activation. This is
the situation in p-n junctions, for instance. A semiconductor diode is thus
commonly used as a noise source by passing a particular DC current through
it. Shot noise is distinct from voltage and
current fluctuations expected in thermal equilibrium; this occurs without any
applied DC voltage or current flowing. These fluctuations are known as
Johnson–Nyquist noise or thermal noise and increase in proportion to the Kelvin
temperature of any resistive component. However both are instances of white
noise and thus cannot be distinguished simply by observing them even though
their origins are quite dissimilar. Since shot noise is a Poisson process
due to the finite charge of an electron, one can compute the root mean square
current fluctuations as being of a magnitude
where q is the elementary charge of an electron, Δf is the bandwidth in hertz
over which the noise is considered, and I is the DC current flowing.
For a current of 100 mA, measuring the current noise over a bandwidth of 1 Hz,
we obtain If this noise current is fed through a
resistor a noise voltage of would be generated. Coupling this noise
through a capacitor, one could supply a noise power of
to a matched load. = Optics=
In optics, shot noise describes the fluctuations of the number of photons
detected due to their occurrence independent of each other. This is
therefore another consequence of discretization, in this case of the
energy in the electromagnetic field in terms of photons. In the case of photon
detection, the relevant process is the random conversion of photons into
photo-electrons for instance, thus leading to a larger effective shot noise
level when using a detector with a quantum efficiency below unity. Only in
an exotic squeezed coherent state can the number of photons measured per unit
time have fluctuations smaller than the square root of the expected number of
photons counted in that period of time. Of course there are other mechanisms of
noise in optical signals which often dwarf the contribution of shot noise.
When these are absent, however, optical detection is said to be "photon noise
limited" as only the shot noise remains. Shot noise is easily observable in the
case of photomultipliers and avalanche photodiodes used in the Geiger mode,
where individual photon detections are observed. However the same noise source
is present with higher light intensities measured by any photo detector, and is
directly measurable when it dominates the noise of the subsequent electronic
amplifier. Just as with other forms of shot noise, the fluctuations in a
photo-current due to shot noise scale as the square-root of the average
intensity: The shot noise of a coherent optical
beam is a fundamental physical phenomenon, reflecting quantum
fluctuations in the electromagnetic field. This sets a lower bound on the
noise introduced by quantum amplifiers which preserve the phase of an optical
signal. See also
Johnson–Nyquist noise or thermal noise 1/f noise
Burst noise Image noise
Quantum efficiency References
This article incorporates public domain material from the General Services
Administration document "Federal Standard 1037C".
External links Shot noise on arxiv.org