Optical Communications, 2nd Edition

5.6 HETERODYNING WITH RANDOM SIGNAL FIELDS

We pointed out that if an optical signal field is transmitted over a turbulent path, the effect is to break up the optical beam spatially. In discussing noncoherent detection in Chapter 4, we considered this effect as an apparent extension of the optical source size. In dealing with coherent detection it is more convenient to consider the turbulent effect as one of converting a coherent optical field to a random field. This randomness is over the receiver spatial variable r and corresponds to a point-to-point loss in coherence over the receiver area. We are here interested in assessing the spatial effects of turbulence on heterodyning. Let us first examine the detected field at a fixed time t. Subsequent time averaging will allow comparison to our earlier SNR results.

Consider the received signal field to be represented as a(t, r) exp (jω_o t) at the receiver aperture, where a(t, r) is again the complex signal envelope. A local heterodyning field source is assumed to produce a single mode diffraction pattern at the photodetector surface. We again describe this local field function over the detector area as a_L exp (jω_Lt). We can again describe the field mixing in terms of the spatial integral over the detector surface. Instead, let us rewrite this heterodyne term in terms of an equivalent integral over the receiver aperture. (Recall that Parcevals theorem allows us to equate an integral in one domain in terms of integrals involving their ...