Spatiotemporal Wave Front Shaping in a Microwave Cavity

Controlling waves in complex media has become a major topic of interest, notably through the concepts of time reversal and wave front shaping. Recently, it was shown that spatial light modulators can counterintuitively focus waves both in space and time through multiple scattering media when illuminated with optical pulses. In this Letter, we transpose the concept to a microwave cavity using flat arrays of electronically tunable resonators. We prove that maximizing the Green’s function between two antennas at a chosen time yields diffraction limited spatiotemporal focusing. Then, changing the photons’ dwell time inside the cavity, we modify the relative distribution of the spatial and temporal degrees of freedom (DOF), and we demonstrate that it has no impact on the field enhancement: wave front shaping makes use of all available DOF, irrespective of their spatial or temporal nature. Our results prove that wave front shaping using simple electronically reconfigurable arrays of reflectors is a viable approach to the spatiotemporal control of microwaves, with potential applications in medical imaging, therapy, telecommunications, radar, or sensing. They also offer new fundamental insights regarding the coupling of spatial and temporal DOF in complex media.

Figure 1

Schematic of the experimental setup, displaying two monopole antennas placed in a reverberant disordered cavity, a spatial microwave modulator (SMM), a mode stirrer, and some electromagnetic absorbers (blue). The emitting antenna is fed by an arbitrary signal generator, while the receiving one is connected to a high sampling rate oscilloscope. The inset (adapted from [46]) illustrates the characteristics of a two-state single SMM element. (a) and (b) show an example of how, using the SMM, we are able to focus the cavity’s Green’s function envelope at a chosen optimization time $t_{\text{opt}}$ (indicated in green). The optimum SMM configuration is identified iteratively with a continuous sequential algorithm. A mode-stirrer rotation by 12° yields a completely uncorrelated, “new” disordered cavity; this is convenient for averaging over many realizations of disorder.

Figure 2

Green’s function envelopes averaged over 60 realizations of the disordered cavities before (thin black line) and after (thick colored lines) temporal focusing of the waves using the SMM and the iterative algorithm on a set of chosen times. The series of experiments has been realized for three different quality factors $Q$ of the cavity. The duration of the focused peaks is always the same: that of the emitted pulse.

Figure 3

Initial (a) and final (b) spatiotemporal maps of the Green’s function envelopes around a given time and position chosen for focusing by wave front shaping, for three different quality factors, and averaged over 30 realizations of disorder. Each temporal wave front shaping experiment results in a spatiotemporally focused wave field onto spots of dimensions ${\lambda }_0/2$ in space and $1/\mathrm{\Delta }{f}_{\mathrm{in}}$ in time.

Figure 4

Quantification of the impact of different cavity dwell times $\tau$ on the enhancement obtained by spatiotemporal wave front shaping at different optimization times $t_{\text{opt}}^{\text{norm}}$ (for clarity in units of the cavity dwell time $\tau$ corresponding to each quality factor). (a) Instantaneous spatiotemporal enhancement ${\eta }_A$. (b) Deposited energy enhancement ${\xi }_E$. All results are based on 60 independent realizations of the disordered cavity. The shaded areas indicate the corresponding uncertainties.