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:

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:

.

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

If we note Δν≃1/2τ the bandwidth of the detector, we can find the useful formula below:

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 the**electric field** applied to the diode.

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

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 a**mean value of zero**. In this case,

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*:

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