The photodetectors used in optical communications are square law detectors. The electrical power they output is proportional to the square of the optical power input.
This is because they generally generate one charge carrier for each photon received. Thus the electrical current is proportional to the optical power. And the electrical power, being proportional to the square of current, is proportional to the square of the optical power.
There are some variations. In any detector, not every incident photon generates a carrier, so the quantum efficiency must be taken into account. In a APD or photomultiplier, the generated carrier triggers an electrical gain mechanism, resulting in an effective response greater than one carrier per photon.
Lets assume we transmit with 20mw optical power, and we receive 10mw in optical power at the photo-diode. What is this 10mw equivalent to in terms of electrical power.
You need to multiply by the responsivity of the detector to get the photocurrent.
$$I = P_o R$$
where $P_o$ is the incident optical power.
The responsivity is given by
$$R = \frac{\eta q}{h \nu}$$
where $\eta$ is the quantum efficiency of the device, $q$ is the electron charge, $h$ is Planck's constant, and $\nu$ is the frequency of the incident light. Generally, $\eta$ will be a frequency-dependent parameter.
To get the electrical power generated, you need to multiply the photocurrent by the potential difference across the detector.
Generally, the electrical power is not a primary concern when analyzing receivers in telecommunications (although it is fundamental to the definition of signal-to-noise ratio). However, solar cells work on the same principle, and there electrical power output is of course the primary goal.
X denotes the electrical signal-to-noise ratio (which is proportional to the square of the received optical power Pr)
This is true when the dominant source of noise in the receiver is from the receiver amplifier (so the noise term is constant and only the signal term varies), which is a very common, but not universal, situation.
