Wave-Field Shaping in Cavities: Waves Trapped in a Box with Controllable Boundaries

Electromagnetic cavities are used in numerous domains of applied and fundamental physics, from microwave ovens and electromagnetic compatibility to masers, quantum electrodynamics (QED), and quantum chaos. The wave fields established in cavities are statically fixed by their geometry, which are usually modified by using mechanical parts like mode stirrers in reverberation chambers or screws in masers and QED. Nevertheless, thanks to integral theorems, tailoring the cavity boundaries theoretically permits us to design at will the wave fields they support. Here, we show in the microwave domain that it is achievable dynamically simply by using electronically tunable metasurfaces that locally modify the boundaries, switching them in real time from Dirichlet to Neumann conditions. We prove that at a high modal density, counterintuitively, it permits us to create wave patterns presenting hot spots of intense energy. We explain and model the physical mechanism underlying the concept, which allows us to find a criterion ensuring that modifying parts of a cavity’s boundaries turn it into a completely different one. We finally prove that this approach even permits us, in the limiting case where the cavity supports only well-separated resonances, to choose the frequencies at which the latter occur.

Figure 1

Concept. (a) Experimental setup [14]: cavity dimensions $1.45\times 1\times 0.75{\mathrm{m}}^3$. (b)–(d) Schematic contribution of the resonances (color lines) to the total transmission (black line) for (b) a large number of modes, (c) the limit of our model, and (d) with $N$ lower than 1. (e)–(g) Schematic view of the random walks of the $N$ contributing modes corresponding to (b)–(d) when the sign of $p$ modes out of $N$ can be controlled. $A_c$ is the sum of the $p$ controlled modes, while $A_u$ is the sum of $N-p$ uncontrolled modes. The total transmission $t$ (in black) corresponds to the sum of $A_c$ with $A_u$. Plain (dashed) arrows: before (after) optimization.

Figure 2

Case (i), $N\gg p$. (a) Typical energy transmission between one emitter and three identical receivers (antennas 1–3), separated by more than a wavelength. Quality factor, $Q=820$; number of SMM elements, $p=102$. The blue (respectively, red) curve is the transmission before (respectively, after) optimization on receiver $R$ (antenna 1). The black and green curves are the transmissions on antennas 2 and 3, respectively, after optimizing on the receiver $R$. (b) The same as (a), but averaged on the stirrer positions and on the initial state of the SMM. (c) Averaged transmission as a function of the iteration number of the algorithm for $Q=820$ and varying $p$. (d) The same for varying $Q$ and keeping $p=102$. (e),(f) Final enhancement $\eta$ versus $p$ for $Q=820$ and versus $Q$ for $p=102$. The theoretical fits correspond to Eq. (7) with $p$ replaced by $\alpha p$ ($\alpha \approx 0.3$).

Figure 3

From case (i) to (ii). (a)–(c) Initial (blue) and optimized (red) transmissions (normalized by the averaged initial transmission) for 700 initial configurations of the SMM for a single cavity, in the complex plane. From (a) to (c), we decrease $N$ from large values ($A_u>A_c$) down to $N/p<1$. (d)–(f) Initial intensity $|t^2|$ probability density function $p_I(I)$ for the three cases. (g)–(i) The same for the probability density function of the phase $p_{\theta }(\theta )$ [note the scale change in (i)].

Figure 4

Case (iii), $N<1$. (a) Experimental setup. Cavity dimensions: $12\mathrm{cm}\times 12\mathrm{cm}\times 24\mathrm{cm}$ with an eight-element SMM. (b) The 2D simulated ergodic cavity of radius $20{\lambda }_0$. SMM of 20 elements. $Q={10}^5$. (c) Experimental transmission $|t|$ at low $N$ for different optimization steps. (d) The same in simulations.