Loss Compensation and Superresolution in Metamaterials with Excitations at Complex Frequencies

Metamaterials, from optics to radio frequencies and acoustics, have attracted significant attention over the last few decades, with promising applications in a wide range of technological areas. However, it has been recognized that their performance is often hindered by ubiquitous material loss and nonlocal phenomena. A canonical problem consists in imaging through metamaterial superlenses, which hold the promise of superresolution but which are, in practice, limited by material loss as we attempt to image deeply subwavelength details. Active metamaterials have been explored to compensate for loss; however, material gain introduces other obstacles, e.g., instabilities, nonlinearity, and noise. Here, we demonstrate that the temporal excitation of passive metamaterials using signals oscillating at complex frequencies can effectively compensate material loss, leading to resolution enhancement when applied to metamaterial superlenses. More broadly, our results demonstrate that virtual gain stemming from tailored forms of excitation can tackle the impact of loss in metamaterials, opening promising avenues for a broad range of applications from acoustic to photonic technologies.

Popular Summary

The resolution of conventional lenses is fundamentally limited by the diffraction of light through the lens. Previously, physicist Pendry proposed that a metamaterial superlens with a negative refractive index could overcome this limitation and achieve perfect imaging. Unfortunately, dissipation due to unavoidable material absorption or scattering losses results in imperfect superlenses. Here, we show that tailoring the temporal excitation of passive metamaterials to synthesize signals oscillating at complex frequencies can compensate for these losses, restoring their superresolution potential.

Generally, metamaterials are artificial materials composed of many unit cells, with geometries engineered to give the overall material properties that do not occur naturally. In this study, we focus on a holey acoustic metamaterial operating as a superlens for sound. This acoustic superlens offers a powerful platform to evaluate the benefits of resolution-boosting metamaterials since it involves a tabletop implementation and it operates at relatively low frequencies, making it easier to control the temporal evolution of the input signal and the corresponding evolution of the images at the focal plane. By properly engineering the excitation signals in time, we show that this device can retrieve deeply subwavelength details of an image that would be otherwise deteriorated by material dissipation. Using a 3D laser Doppler vibrometer, we show how complex frequency excitations can restore the resolution of the metamaterial device in real time.

Beyond enhancing the resolution of a practical imaging system, our results demonstrate that complex frequency excitations can be used to compensate for material loss in metamaterial devices, paving the way to overcome many of the challenges in translating exciting theoretical proposals into practical devices for various applications.

Figure 1

Superlens response limited by material loss. (a) Superlens without loss, amplifying evanescent fields and providing perfect imaging of an object at the image plane. (b) Material loss in the superlens, which is detrimental to the amplification of evanescent waves, resulting in limited resolution. (c) Excitation at a suitable complex frequency, imparting virtual gain to a passive lossy superlens and enabling an enhancement in resolution and compensation of the detrimental effect of material loss.

Figure 2

Virtual gain compensating material absorption in a Drude superlens. (a) Transmission versus transverse wave number for various decay rates $\gamma =0$ (harmonic), $0.75\alpha$, $0.9\alpha$, and $\alpha$. The impinging waves oscillate at the same real part of ${\omega }_c$ with different decay rates. At $\gamma =\alpha$ (yellow curve), the dispersion becomes identical to the dispersion of the ideal. (b) Normalized transmission $|\tilde{T}(k_{x0},\omega )|$ for $k_{x0}=30k_0$. The transmission is unitary at complex frequency ${\omega }_c=5.77\times {10}^{15}-i5\times {10}^{14}\mathrm{rad}/\mathrm{s}$. (c) Density map of the transmission as a function of decay rate and $k_x$, confirming that transmission is unity when the decay rate is equal to the intrinsic loss rate of the system. (d) Point spread function in the image plane for $\gamma =0$, $0.75\alpha$, $0.9\alpha$, and $\alpha$. As the decay rate approaches the loss rate, the function becomes close to the lossless scenario (dotted curve). The width of the function at $\gamma =\alpha$ is about 5 times smaller than the width of the function at $\gamma =0$.

Figure 3

Holey acoustic metamaterial under complex frequency excitation. (a) Schematic of the acoustic metamaterial with thickness $h=10\mathrm{cm}$. The size of the square holes is $a=0.15\mathrm{cm}$, and the lattice constant is $d=0.2\mathrm{cm}$. (b, left panel) Transmission versus transverse wave number. The transmission at the first FP resonance is not unity for large $k_x$ (yellow) due to nonlocality, such that its response is fundamentally limited by $2\pi /d\approx 100k_0$ (red dotted line). In the lossy case, the dispersion (red) deviates from the lossless case (yellow). (b, right panel) Incident waves impinging on the structure at the first FP resonant frequency with different complex frequencies. For $\gamma =\alpha$, the dispersion is identical to the dispersion of the lossless scenario (yellow in the left panel). (c) Schematic of the object plate with three lines. The width is 0.4 cm, and the gaps are 0.2 and 0.4 cm. (d) Sound intensities for various decay rates $\gamma =0$, $0.5\alpha$, and $\alpha$. The dotted boxes represent the original three-line object. The image at $\gamma =\alpha$ becomes consistent with the image in the harmonic lossless case. (e) Image cut of sound intensities for each case. (f) Two-dimensional spatial Fourier components of the sound maps in Fig. 3 (log scale).

Figure 4

Experimental demonstration of virtual acoustic superlens. (a) Three-line object, sound maps in real and reciprocal spaces for each case. The maps represent acoustic energy normalized by each maximum. Once the decay rate is close to the loss rate $\alpha \approx 180\mathrm{Hz}$, the image map is optimally resolved. (b) One-dimensional Fourier components for different decay rates normalized to the same transverse wave number in the harmonic excitation case. (c) Comparison of letter E (top) and smiley face (bottom) images under harmonic and complex frequency excitations. Here, the decay rates of the complex excitation are 90 and 180 Hz.