Noise in photodetectors (Shot noise)

After my recent post on power and energy meters, I’d like to speak about the different kind of noises that affect a photodetector and how to model them.

Quantum effect and noise: the shot noise.

Fundamental physics tells us the light is made of particles (photons), which are emitted by the source at random. For that reason, the amount of photons emitted by the source (sun, bulb, laser, etc.) is not constant, but exhibits detectable statistical fluctuations. And this is in a nutshell what shot noise is. Because of its nature, it does not depends on the quality of the detector and is unavoidable. However the shot noise becomes a real issue only when the optical intensity is fairly low: in this case quantum fluctuations become much more noticeable.

The random process of light emission can generally be modelled using a Poisson distribution, the properties of which are very well known. If we note p(n) the probability that n photons arrive on the detector:

p(n)=n¯nexp(n¯)n!
n¯=0+np(n)
σ2n=0+(nn¯)2p(n)=n2¯¯¯¯n¯2=n¯

where σn is the standard deviation. What this means is that for 100 photons arriving on the detector, the uncertainty about the number of photon is of ±10 (±10%). If the number of photon is somewhat closer to common levels, e.g. 1010, the uncertainty becomes ±105, which is ±0.000,01%. It then becomes obvious that the shot noise is an issue only at low light level.

Let’s find out now what would be the fluctuations of the signal current due to the shot noise. Since each photon induce a free electron with an efficiency η, during the time τ the number of electron produced is then ηn¯. Every electron contributes to the signal current is for a charge e, the average value of the signal current is¯¯¯ is then:

is¯¯¯=eτηn¯

.

Fluctuation in the number of photons create a fluctuation in the signal current, and those fluctuations are characterised by the standard deviation:

σ2is=(eτ)2σ2ηn=(eτ)2ηn¯

If we note Δν1/2τ the bandwidth of the detector, we can find the useful formula below:

σ2is=2eis¯¯¯Δν

And this standard deviation σis characterise the shot noise current.

Dark current

The dark current is the constant response exhibited by a detector during periods when it is not actively being exposed to light. It is sometime classified as another type of shot noise. It is also referred to as reverse bias leakage current in non optical devices and is present in all diodes. Physically, dark current is due to the random generation of electrons and holes within the depletion region of the device that are then swept by theelectric field applied to the diode.

Similarly to the photon noise, it can be modelled by a Poisson distribution, with:

σ2i0=2ei0¯¯¯Δν

Let’s note that i0¯¯¯ depends from many parameters, but generally speaking:

  • Si photodiodes : i0¯¯¯ ranges from 1 to 10 nA
  • Ge photodiodes: i0¯¯¯ ranges from 50 to 500 nA
  • InGaAs photodiodes: i0¯¯¯ ranges from 1 to 20 nA

Thermal noise in photodetectors

Thermal noise, also called Johnson noise or Nyquist noise is the electronic noise generated by the thermal agitation of the electrons inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. It was discovered by Johnson in 1927 and explained by Nyquist.

A device (a photodiode for instance) thermal noise can be modelled as a voltage source Vth(t) in series with an ideal resistor R. Vth(t) has a Gaussian distribution with amean value of zero. In this case,

σ2Vth(t)=4kBTRΔν

Where σVth(t) is the standard deviation of the voltage, kB is the Boltzmann’s constant in joules per kelvin and T is the resistor’s absolute temperature in kelvins. It can also be modelled a current source ith in parallel with R:

σ2ith(t)=4kBTRΔν

3 thoughts on “Noise in photodetectors (Shot noise)

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