Geometric loss calculation

Geometric loss is the loss caused by the inevitable spreading of the signal beam as it travels through space down to earth. Although the beam is also spread (probabilistically) by pointing errors and atmospheric turbulence, these contributions are considered separately.

Gaussian beam

Geometric loss can be calculated using the normalised beam profile (the beam profile with unity power). For a gaussian beam, this is described as

$$f(\underline{r}) = \frac{1}{2\pi w^2} exp(-\frac{|\underline{r}|}{2w^2})$$

Where $w = \frac{FOV_t \ l}{2} + \frac{d_t}{2}$ is the beam half-width at the receiver, depending on the transmitter field of view $FOV_t$, the link range $l$ and transmitter telescope diameter $d_t$.

For a long (satellite) link, it is reasonable to assume that the receiver telescope diameter $d_r$ is much smaller than the beam width at the receiver $w$. In this case, it is valid to approximate the intensity at the receiver by a constant value. For geometric loss, we assume pointing is perfect, so we can use

$$Geometric \ Loss \approx f(\underline{0})\times A_{receiver} = \frac{1}{2\pi w^2} \ \pi \left(\frac{d_t}{2}\right)^2$$

Flat-top beam

A flat-top beam is a simpler case, with the normalised intensity profile simply being uniform over a circle with half-width $w$.

$$f(\underline{r}) = \frac{1}{\pi w^2}$$

So that geometric loss is given by

$$Geometric \ Loss \approx f(\underline{0})\times A_{receiver} = \frac{\pi \left(\frac{d_r}{2}\right)^2}{\pi w^2}$$