Photon statistics describe how likely it is to detect different photon numbers in a light field.
Even when two light sources have the same average intensity, their photon-number distributions can be very different.
This is one of the most important ideas in quantum optics because the distribution reveals information about the physical nature of the source.
In this calculator, the two basic cases are coherent light and thermal light.
Ideal coherent light, such as an ideal single-mode laser far above threshold, follows a Poisson distribution.
If the mean photon number is
\(\mu=\langle n\rangle\),
then the probability of detecting exactly \(n\) photons is
\[
P_{\rm P}(n)=e^{-\mu}\frac{\mu^n}{n!}.
\]
A defining feature of Poissonian light is that its variance equals its mean:
\[
\mathrm{Var}_{\rm P}(n)=\mu.
\]
This means the fluctuations grow with the signal level, but only in a very specific way.
The standard deviation is
\(\sqrt{\mu}\),
so the relative fluctuations become smaller as the mean photon number becomes larger.
That is one reason coherent laser light is often viewed as comparatively “quiet.”
Thermal light, also called chaotic light, follows a very different photon-number distribution.
For a single thermal mode with the same mean photon number \(\mu\), the probability is
\[
P_{\rm th}(n)=\frac{\mu^n}{(1+\mu)^{n+1}}.
\]
This distribution is broader and more strongly concentrated near small photon numbers.
Its variance is
\[
\mathrm{Var}_{\rm th}(n)=\mu(\mu+1).
\]
Since
\(\mu(\mu+1)>\mu\)
whenever
\(\mu>0\),
thermal light fluctuates more than coherent light at the same mean intensity.
This extra noise is one of the hallmarks of chaotic radiation.
A useful way to compare the fluctuation strength is the Fano factor,
\[
F=\frac{\mathrm{Var}(n)}{\mu}.
\]
For Poissonian light,
\[
F_{\rm P}=1,
\]
while for thermal light,
\[
F_{\rm th}=\frac{\mu(\mu+1)}{\mu}=\mu+1.
\]
So thermal light is super-Poissonian, meaning its fluctuations are larger than the Poisson baseline.
Coherent light is exactly Poissonian.
In more advanced quantum optics, one also studies sub-Poissonian light, such as certain squeezed or antibunched states, for which the fluctuations are even smaller than Poissonian.
Consider the sample case
\(\mu=\langle n\rangle=5\).
For coherent light, the variance is immediately
\[
\mathrm{Var}_{\rm P}=5.
\]
For thermal light with the same mean,
\[
\mathrm{Var}_{\rm th}=5(5+1)=30.
\]
So the thermal distribution is much wider.
This is exactly what the histogram comparison in the calculator is designed to show: even when the average photon number is identical, the thermal distribution spreads over a much larger range of photon numbers.
Another intuitive difference appears at
\(n=0\).
For coherent light,
\(P_{\rm P}(0)=e^{-\mu}\),
while for thermal light,
\(P_{\rm th}(0)=1/(1+\mu)\).
For moderate mean photon numbers, the thermal source can still have a surprisingly large probability of yielding very low photon counts, while the coherent source is more concentrated around the mean.
At a more advanced university level, photon statistics connect to correlation functions such as
\(g^{(2)}(0)\),
shot noise, antibunching, squeezed states, and nonclassical light.
Coherent light has
\(g^{(2)}(0)=1\),
thermal light has
\(g^{(2)}(0)=2\),
and antibunched light can satisfy
\(g^{(2)}(0)<1\).
But for a first comparison, the mean, variance, and histogram already reveal the essential physics.
\[
P_{\rm P}(n)=e^{-\mu}\frac{\mu^n}{n!},
\qquad
\mathrm{Var}_{\rm P}=\mu,
\]
\[
P_{\rm th}(n)=\frac{\mu^n}{(1+\mu)^{n+1}},
\qquad
\mathrm{Var}_{\rm th}=\mu(\mu+1).
\]
These formulas explain why ideal laser light and thermal light can have the same average brightness but very different statistical noise properties.
