Shot Noise

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Shot noise is a term describing the random fluctuations in a measurement signal due to the random arrival time of the signal carriers (electron, photon, etc.). The process for an electrical current is first described, then related to noise spectra and optical shot noise.

Poisson process for electrons

Consider the electrical current $I (t)$ associated with a fixed rate, $K$ , of electron arrival at some measurement point. The average current is given by

$\langle I \rangle = K e$

where $e$ is the electron charge. If we assume that electron arrival events are uncorrelated from one another, then the current will be described by a Poisson Process. For a time interval $τ$ , with $n$ electron measurement events. The average number of these events is then

$\langle n\rangle=K\tau$

and the variance in $n$ is

$\langle n -\bar{n}\rangle^2$	$= \langle n^2\rangle-\langle n\rangle^2$
	$= \langle n\rangle = K \tau \ ,$

since the process is Poissonian. It follows that the mean square value of the current is given by

$\langle I^2\rangle = \left\langle\left(\frac{ne}{\tau}\right)^2\right\rangle = \left(\frac{e}{\tau}\right)^2 \langle n^2\rangle \ .$

Since

$\langle n^2\rangle$	$= \left(\langle n^2 \rangle-\langle n\rangle^2\right) + \langle n\rangle^2$
	$= \langle n\rangle + \langle n\rangle^2$

we arrive immediately at

$\langle \Delta I^2\rangle = \langle I^2\rangle -\langle I\rangle^2 = \frac{\langle n\rangle e^2}{\tau^2} = \frac{\langle I\rangle e}{\tau} \$

Noise spectra and filtering

Now we will derive the noise spectrum of the current $S i (f)$ such that the rms current can be represented as

$i_{rms}^2=\int_0^\infty |\mathcal{G}(f)|^2 S_i(f) df$

For a white noise floor that is independent of frequency: $S i (f) = S i$ we have

$i_{rms}^2=S_i \int_0^\infty |\mathcal{G}(f)|^2 df = S_i \Delta f$

where we have defined

$\Delta f=\int_0^\infty |\mathcal{G}(f)|^2 df$

Example filter responses

Windowing filter

If we accept all frequencies only between $f 1$ and $f 2$ then we have trivially

$Δ f = f 2 - f 1$

Exponential decay filter

For an exponential decay response with time-constant $τ$ we have

$\Delta f=\int_0^\infty \frac{df}{1+(2\pi f \tau)^2}=\frac{1}{4\tau}$

Single pole filter

For a single pole filter with 3dB frequency $f c$ we have

$\Delta f=\int_0^\infty \frac{df}{1+(f/f_c)^2}=\frac{\pi f_c}{2}$

Averaging filter

The response function for averaging for a time $τ$ is given by

$\Delta f=\int_0^\infty \left(\frac{\sin(\pi f \tau)}{\pi f \tau}\right)^2 df=\frac{1}{2\tau}$

This can be derived from the fourier transform of the windowed averaging filter:

$\mathcal{G}(f)=\int_{-T/2}^{T/2} \frac{\exp(-i 2\pi f t )}{T} dt = \frac{\sin(\pi f T)}{\pi f T}$

Shot noise spectrum

Now returning to the above shotnoise expression and using the fact that an averaging filter is used to give $Δ f = 1 / 2τ$ along with the definitions $i_{rms}^2=\langle \Delta I^2\rangle$ and $I = \langle I \rangle$ we have

$i_{rms}^2= \frac{e I}{\tau} = 2 e I \Delta f$

and

$i_{rms} = \sqrt{2 e I} \left[\textrm{A/}\sqrt{\textrm{Hz}}\right] \sqrt{\Delta f}$

Now we can use this expression independently of the averaging filter that was used in the derivation and apply any frequency filter.

Converting to optical shot noise

To derive the noise seen in a photodetector, one simply has to use the conversion factor

I = R P

where $I$ is the electrical current as before, $P$ is the optical power,

$R = \frac{\eta e}{\hbar \omega} \left[\textrm{A/}\textrm{W}\right]$

is the responsivity of the detector, and $η$ is the quantum efficiency by which the detector converts a photon (of energy $\hbar \omega$ ) into a current carrying charge, $e$ .