Geometric loss calculation

Geometric Loss

In principle, geometric loss represents the spreading of an optical beam over a channel, resulting in not all of the transmitted light being incident at the detector. The conceptual relationship underpinning geometric loss is

$$Loss = \frac{1}{transmitted \ power}\int_{receiver \ area} transmitter \ intensity(\mathbf{x}) dA$$

This integral must be simplified to be tractable in simulation. Luckily, for common optical arrangements, these simplifications exist.

Flat-top beams

Flat-top beams have uniform intensity over a well-defined (in our case, circular) beam area. If the beam area includes the receiver (which it should, see "Acquisition, pointing and tracking (APT) loss calculation"). Then the above integral simplifies to

$$Loss = \frac{receiver \ area}{beam \ area}$$

In turn, the beam area can be related to the link range $l$, the beam divergence angle $\omega_{FT}$ and the transmitter telescope diameter $T$ by noting that $beam \ diameter \ at \ receiver = T + \omega_{FT} l$. The receiver diameter $D$ drives the receiver area in a similar way.

$$Loss = \frac{D^2}{(T + \omega_{FT} l)^2}$$

Gaussian Beams

Gaussian beams have an intensity profile which is nonzero everywhere (although clearly this is only ever approximated in reality). Nonetheless, this means that the concept of beam area has less meaning. Instead, we have only a measure of how 'spread' the beam is, $\omega_{G}$, the angle at which intensity decays to $\frac{1}{e^2}$ its maximum value.

The derivation of the result for the geometric loss of a gaussian beam is performed in more detail in "Acquisition, pointing and tracking (APT) loss calculation". We use the result

$$Loss = \frac{\sqrt{\pi}}{8}\frac{D^2}{(T + \omega_{G} l)^2}$$