Communication via Holomorphic Green Functions

Let G(xr − xe) be the causal Green function for the wave equation in four spacetime dimensions, representing the signal received at the spacetime point xr due to an impulse emitted at the spacetime point xe. Such emission and reception processes are highly idealized, since no signal can be emitted or received at a single (mathematical) point in space and time. We present a simple model for extended emitters and receivers by continuing G analytically to a function G̃(zr − ze), where ze = xe + iye is a complex spacetime point representing a circular pulsed-beam emitting antenna dish centered at xe and emitting in the direction of ye, and zr = xr − iyr represents a circular pulsed-beam receiving antenna dish centered at xr and receiving from the direction of yr. The holomorphic Green function G̃(zr − ze) represents the coupling between the emission from ze and the reception at zr. To preserve causality and give nonsingular coupling, the orientation vectors ye and yr must belong to the future cone V+ in spacetime. Equivalently, ze and zr belong to the future and past tubes in complex spacetime, respectively. The space coordinates of ye and yr give the spatial orientations and radii of the dishes, while their time coordinates determine the duration and focus of the emission and reception processes. The directivity D(y) of the communication process is a convex function on V+, i.e., D(yr + ye) ≤ D(yr) +D(yr). This shows that the efficiency of the communication can be no better than the sum of its emission and reception components. Supported by AFOSR Grant #F49620-01-1-0271. February 7, 2008