Green’s Function Retrieval with Absorbing Probes in Reverberating Cavities

The cross-correlation of a diffuse wave field converges toward the difference between the anticausal and causal Green’s functions between two points. This property has paved the way to passive imaging using ambient noise sources. In this Letter, we investigate Green’s function retrieval in electromagnetism. Using a model based on the fluctuation dissipation theorem, we demonstrate theoretically that the cross-correlation function strongly depends on the absorption properties of the receivers. This is confirmed in measurements within a reverberation chamber. In contrast to measurements with noninvasive probes, we show that only the anticausal Green’s function can be retrieved with a matched antenna. Finally, we interpret this result as an equivalent time-reversal experiment with an electromagnetic sink.

Figure 1

Normalized cross-correlation $C(t)$ obtained from measurements of the field with a noninvasive electro-optical probe translated at two positions.

Figure 2

(a) Normalized cross-correlation $C(t)$ for a horn antenna and a noninvasive probe. (b) Ratio of the maximum amplitude at negative times and maximum amplitude at positive times with the impedance mismatch $\mathrm{\Gamma }$ for an ultra-wideband horn antenna (star), two other horn antennas (circle and cross), a discone antenna (square), and a log-periodic dipole antenna (triangle). The straight line is $R=\mathrm{\Gamma }$. (c) Normalized cross-correlation $C(t)$ for the same receiving horn antennas facing each other.

Figure 3

(a) Color map representation of $C(t)$ when the distance between the horn antenna and the probe increases from 0.55 m to 1 m. (b) $dC(t)/dt$ (blue) taken at negative times is compared to the impulse response (red).

Figure 4

The time-reversed signal is plotted for three load impedances: a $50\mathrm{\Omega }$ load (blue), an open-circuit load (red), and a short-circuit load (black).