Optical Communications, 2nd Edition

2.2 POWER DETECTION AND RECEIVER FIELD OF VIEW

In the optical receiver using lenses, the aperture field is imaged on to the focal plane, where the photodetector is placed. The photodetector, therefore, responds to the imaged field on its collecting surface. Two important parameters are the amount of power in the detector field and the receiver field of view.

Detector field power can be determined by direct application of Parceval's theorem in two-dimensional transform theory. Let f₁(x, y) ↔ F₁(u, ν) and f₂(x, y) ↔ F₂(u, ν) be two transform pairs. Then Parceval's theorem states

When f₁ = f₂, this reduces to

This can be related to the integrals of the aperture and to the focused fields of an optical lens, which are themselves transforms. In particular, it can be stated that a direct relationship exists between spatial integrals of focused fields after lens transformations. Let us apply the field power definitions in Eq. (1.3.3) with

Then, by use of Eq. (2.1.4), the integral in Eq. (2.2.3) becomes

where F_r(t, ω₁, ω₂) is the inverse Fourier transform of f_r(t, x, y) over the aperture area ...