Exploiting spatiotemporal degrees of freedom for far-field subwavelength focusing using time reversal in fractals

Materials which possess a high local density of states varying at a subwavelength scale theoretically permit the focusing of waves onto focal spots much smaller than the free space wavelength. To do so, metamaterials—manmade composite media exhibiting properties not available in nature—are usually considered. However, this approach is limited to narrow bandwidths due to their resonant nature. Here, we prove that it is possible to use a fractal resonator alongside time reversal to focus microwaves onto $\lambda /15$ subwavelength focal spots from the far field, on extremely wide bandwidths. We first numerically prove that this approach can be realized using a multiple-channel time reversal mirror that utilizes all the degrees of freedom offered by the fractal resonator. Then, we experimentally demonstrate that this approach can be drastically simplified by coupling the fractal resonator to a complex medium, here a cavity, that efficiently converts its spatial degrees of freedom into temporal ones. This makes it possible to achieve deep subwavelength focusing of microwave radiation by time reversing a single channel. Our method can be generalized to other systems coupling complex media and fractal resonators.

Figure 1

Focusing with a Hilbert resonator. (a) Experimental setup: A network analyzer measures the transmission of a metallic Hilbert curve inserted in a metallic screen between a horn antenna and a near-field probe. (b) Transmission spectrum averaged on the positions. (c) Typical transmission signal at ${\mathbf{r}}_1=(34\text{mm},59\text{mm})$. (d)–(f) Time reversal focusing (maximum over time of the energy) at positions ${\mathbf{r}}_1,{\mathbf{r}}_2=(73,70)$, and ${\mathbf{r}}_3=(66,35)$. Red and yellow curves give the profiles on the dashed white lines whose intersections show the focal spots.

Figure 2

Simulations: Focusing with an order 6 Hilbert fractal and ten spatial degrees of freedom. (a) Simulated setup. The Hilbert curve is illuminated by 90 plane waves (45 angles of incidence and two orthogonal polarizations) of Gaussian pulses; three of them are represented in red, blue, and green. The electric field is recorded in plane parallel to the fractal at a distance of 1 mm. (b)–(d) Time maximum value of the energy when focusing at positions ${\mathbf{r}}_1,{\mathbf{r}}_2$, and ${\mathbf{r}}_3$.

Figure 3

Focusing with a Hilbert fractal and a cavity. (a) Experimental setup: A network analyzer measures the transmission of a metallic Hilbert fractal inserted in the wall of a high $Q$ cavity, between an inside isotropic antenna and a near-field probe. (b) Schematic view of the increase of degrees of freedom: black, real cavity; red, image sources and cavities. (c) Time signal in transmission at ${\mathbf{r}}_1$. (d) Transmission spectrum averaged on the positions. (e)–(g) Time reversal focusing with a cavity at positions ${\mathbf{r}}_1,{\mathbf{r}}_2$, and ${\mathbf{r}}_3$. (h) Average size (100 random positions), and standard deviation (error bars) of the focal spots obtained with fractals of order 4 to 7 with the usual dielectric substrate and with an order 6 made with a low loss substrate.

Figure 4

Comparing the spatial coherence. (a) Measured transmission spectrum of a Hilbert fractal of order 6 without and with a cavity between 2.18 and 2.28 GHz. The green dashed arrows refer to the modes picked up for (d) and (g). (b), (c) Phase of the modes 1 and 10 at 2.18 and 2.23 GHz without a cavity. (d) Spatial cross correlations of the modes measured without a cavity. (e)–(g) Same as (b)–(d) with a cavity.