Highly-Efficient Focusing of Terahertz Waves with an Ultra-Thin Superoscillatory Metalens: Experimental Demonstration

: The performance of an ultra-thin ( thickness < 0.04 λ 0 ) metasurface superoscillatory lens (metaSOL) is experimentally demonstrated in the terahertz (THz) range. The metaSOL is designed using two different hexagonal unit cells to improve the efficiency and properties of the conventional transparent-opaque zoning approach. The focusing metastructure produces, at a frequency f exp = GHz, a sharp focal spot 8.9 λ exp away from its output surface with a transversal resolution of 0.52 λ exp (≈ 25% below the resolution limit imposed by diffraction), a power enhancement of 18.2 dB and very low sidelobe level (  13 dB). Resolution below the diffraction limit demonstrated in a broad 18%. proposed metaSOL potential a of such as imaging,


Introduction
Metamaterials and their 2D version, metasurfaces, have attracted much attention owing to the possibility they offer to control phase, amplitude and polarization characteristics of electromagnetic (EM) waves. This flexibility opens new avenues to engineer complex structures able to surpass the limits imposed by natural materials. [1] In fact, it is possible to synthesize artificial materials with precise electric and magnetic responses and thus obtain exotic EM features such as near-zero and/or negative refractive index values. [2][3][4][5][6] Metamaterials and metasurfaces have been applied in multiple scenarios such as lenses, sensors, antennas, circuits and computing, among others, along with spatio-temporal modulation of their EM properties, for frequencies from acoustics to the optical range. [7][8][9][10][11][12][13][14][15][16] These man-made materials have also found applications in the emerging THz range (0.1 to 10 THz) in a wide variety of fields such as sensing, spectroscopy and imaging, defense and security, material testing, biomedicine, to name a few. [17][18][19][20][21][22][23] Imaging devices usually suffer from the diffraction produced by EM waves, which prevents resolving subwavelength features. It is well known that the spatial resolution of conventional lenses (such as the commonly used dielectric lenses and parabolic mirrors) is in the order of 0.61λ0 ∕NA, [24] where NA is the numerical aperture of the lens and λ0 is the free space operation wavelength.
Overcoming this diffraction limit has become a hot research topic worldwide and different ways to surpass it have been proposed, such as: (i) photonic nanojets for near-field imaging using dielectric microspheres and cuboids, [25][26][27] (ii) near-field optical-scanning microscopy systems, which are able to detect the high frequency components from evanescent waves using probes located within less than one wavelength from the sample surface, allowing resolutions of λ0/20 for two-dimensional objects; [28,29] (iii) high-refractive index glass microspheres combined with a conventional The lens analyzed in this study is based on the design theory and methods presented in a previous article published by some of the authors of this work, [38] and has been conceived to overcome the challenge of balancing the trade-off between the different parameters involved. Thus, here we demonstrate experimentally a metaSOL designed and operating in the lower frequency band of the THz spectrum (300 GHz) to: i) Enhance the transmission efficiency above 70%, improving the 50% transmission efficiency achieved with the typical transparent and opaque ring masks configuration [43][44][45][46] and, all the more, the 5-10% transmission efficiency of other SOL design methods such as hole arrays [47] and superlenses [48] ; ii) enhance the focusing efficiency, also known as the yield parameter, defined as the ratio of the energy located in the hotspot to the full focal plane (see supplementary material for a more detailed explanation of this parameter). A yield above 50% will be imposed by design, because the energy at the focus determines the limit of how small the hotspot can be while remaining effective in imaging. [47] This will affect the size of the hotspot, since even for the largest-sized SOLs (with a diameter of the order of 10 3 wavelengths) the focusing efficiency reported is only 5% at most when the hotspot is well below the diffraction limit (resolution ≈ 0.38λ0 ∕NA); [49] iii) obtain a focal spot size beyond the diffraction limit with small sidelobes to get sub-wavelength focusing with a large FoV. The focus size is usually evaluated by means of its full width at half maximum (FWHM), defined as the distance at which the power distribution has been reduced to half its maximum at the focal spot whereas the sidelobe level (SLL) is defined as the ratio between the maximum power at the focal point and the maximum power of the first sidelobe (more details about these parameters can be found in the supplementary material); iv) obtain an ultra-thin and compact lens, with a thickness < 0.04λ0 and diameter < 55λ0, the metaSOL becomes 1 to 2 orders of magnitude electrically smaller than the SOLs designed for the optical spectrum [48][49][50] and up to 3 times smaller than the SOLs reported in the THz range [37] .
As a trade-off between the focal point intensity, size and FoV, the engineered metaSOL achieves a transmission and focusing efficiency of ≈71% and ≈73% respectively, with FWHM = 0.52λexp, SLL

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THz spectrum (

Lens design and fabrication
Although the design process was fully explained in [38] , here we summarize it for the sake of clarity and completeness. Superoscillation is a phenomenon in which a signal that is globally band-limited can contain local ultra-high spatial frequencies that oscillate faster than its maximum Fourier components. This property enables its use for far-field super-resolution focusing. [51][52][53][54][55][56][57][58] Such a superoscillatory function can be easily obtained by the superposition of wave functions.
Following this reasoning and applying the Huygens-Fresnel (H-F) principle, [59] each unit cell in our design is analytically considered as a point source that radiates a cylindrical wave whose amplitude and phase is equal to its transmission coefficient (tn), assuming that the excitation is done with a plane wave of unit amplitude at normal incidence. The resulting field at each point of space can be calculated by adding the fields of all sources. Mathematically, this can be written as: where f(r) is the electric field at the evaluation point; N is the number of source points; r and r' are the position vectors at the evaluation point and at the source point, respectively; k0 is the wave number in free space (assuming that the lens in embedded in free space); |tn| and arg{tn} are the transmission coefficient magnitude and phase at each point, respectively.
By analogy to other superoscillatory functions [60,61,62] the H-F principle provides at the focal plane a resulting field that can show a superoscillatory behavior if the transmission coefficient magnitude and phase at each point of the lens is adequately modulated. As it is known, a superoscillatory function can be obtained by the linear superposition of wave functions, enabling the H-F model, even despite its simplicity, to develop a sub-wavelength superoscillatory focus. (1)

Lens design
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Lens design
Although the design process

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Although the design process and completeness

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By analogy to other superoscillatory functions

By analogy to other superoscillatory functions
This article is protected by copyright. All rights reserved The metaSOL is then analytically engineered through a modified algorithm based on the combination of a genetic method, the Binary Particle Swarm Optimization (BPSO) [63] and the H-F principle. For this purpose, two hexagonal unit cells that cover, respectively, the even and odd zones, are designed to provide a high transmission coefficient magnitude (|tn| = 0.8) while ensuring enough phase difference between them (|arg{tn,odd} arg{tn,even}| > 0.16π rad) at the operation frequency (fixed in the design at f0 = 327 GHz, λ0 = 0.917 mm), so the algorithm can converge and generate the focal spot at the desired FL (fixed in the design at 10λ0). As demonstrated in our previous work, [38] unlike the usual lenses based on binary phase masks, where the phase difference is around π, our design enabled us to use a much smaller phase difference and to benefit from a high transmission magnitude in both cells. We performed a preliminary analytical study using the H-F principle and found that a phase difference of at least ~0.16π (π/6) rad between even and odd zones was necessary for the BPSO algorithm to converge adequately and generate the focal spot. A complete discussion can be found in [38] The full metaSOL and the designed unit cells are represented in Figure 1 (a,b), respectively. As in [38] , we first evaluated a cylindrical lens (see Figure 1 (c)) via the H-F approximation. Then, the final spherical lens shown in Figure 1(a) was obtained by applying rotation symmetry to the cylindrical lens solution, and thus obtaining radial zones with 18783 unit cells covering a circular area of diameter  = 49.7 mm (54.20) and thickness Lz = 0.036 mm (0.0390). It must be noted that this evolution from the cylindrical to the spherical lens affected the original FL, from 10λ0 to 9.2λ0.

Focusing properties
The fabricated metaSOL was experimentally measured using the planar near-field characterization technique (detailed below) and then compared with analytical (H-F method) and simulation (CST Studio Suite ® ) results. In the experimental setup the lens was illuminated with a standard horn antenna with a moderate directivity (20 dBi) that generated a y-polarized (see coordinate axes in Figure 1(a)) Gaussian beam with a low ellipticity ratio. The source was placed 110 mm away from the lens plane, providing non-uniform illumination, so that the difference (in decibel scale) between the electric field intensity at the central point of the lens and its edges was around 10 dB, to minimize the spillover and hence the direct transmitter-to-receiver transmission. As a receiver, an

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This article is protected by copyright. All rights reserved reduce reflections. Absorbent material was used to reduce the reflections from the UG/387-U flange.
Although the probe suffers from some reflections from the tip that cannot be avoided, they are sufficiently small to be neglected. Regarding the crosspolar isolation, this probe presents at least 30 dB isolation to the orthogonal polarization. Millimetre-wave extenders (VDI WR3.4-VNAX) were connected to both ports of an Agilent N5242A PNA-X Network Analyzer and the receiver probe was fixed on a motorized 2-axes translation stage. The experimental setup was placed on a planar antivibration table to minimize noise from mechanical vibrations and both the metaSOL and receiver probe were surrounded with millimetre-wave absorbing material to minimize reflections and mimic anechoic chamber conditions. Also, a wide time-domain gating was set on the Vector Network Analyzer (VNA) to reduce the effect of parasitic reflections.
Before starting the measurement process, the whole system was calibrated by placing face to face the transmitter and receiver at a distance z = 9 mm without the metaSOL and sweeping the frequency from 260 to 350 GHz with a step ∆f = 1 GHz (91 frequency points). To verify that using this single measurement in the optical axis for each frequency was sufficient as a calibration, the magnitude of the electric field was also recorded in the interval from z = 6 to z = 14 mm, without the metaSOL. The ripple signal obtained by comparing the new measurements at the different z positions to the reference signal (z = 9 mm) was limited to be ±0.75 dB in the whole frequency range. This small variability supports the simplification assumed, necessary to speed up the measurements.
After calibration, the metaSOL was inserted in the experimental setup. The electric field (E-field) amplitude and phase was recorded by scanning 5×5mm xy-plane squares, with steps of ∆x = ∆y = 0.1 mm (i.e., 51×51 points) at distances from z = 6 mm to z = 14 mm away from the aperture plane

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After calibration, the metaSOL was inserted in the experimental setup.

amplitude and phase
Accepted Article amplitude and phase was Accepted Article was extract them from the experimental data is shown in the supplementary material section. In Figure   2, we represent the analytical and experimental power distribution spectral maps when the probe is moved only along the optical z-axis (x = y = 0).

Figure 3. (a) Experimental power enhancement (in decibel scale) along the frequency range 275-332 GHz at the corresponding FL for each frequency. (b) Average experimental FWHM (red curve) and Rayleigh diffraction limit (blue curve) both normalized to λ.
The average FWHM, calculated by taking the average FWHM along radial directions around the focus, is represented in Figure 3b (red curve) and compared to the Rayleigh diffraction limit (blue curve), both normalized to the wavelength. At fexp the FWHM is 0.53 mm (0.52exp) whereas the Rayleigh diffraction limit is 0.66 mm (0.65exp). Note that each frequency has a different FL (as can be seen in Figure 2b) due to chromatic aberration and hence, the FWHM calculation is done at a different z position for each frequency. Since all the foci obtained are below the diffraction limit at the corresponding FL for each frequency, this leads to a high fractional bandwidth around 18%.
To fully characterize the focal image space of the lens at the working frequency (f0 in the analytical and simulation results and fexp in the experimental results), additional transmittance maps were obtained. The yz-maps at x = 0 mm and xz-maps at y = 0 mm, are shown in Figure 4 (panels a-c and e-g).

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. Normalized power distribution at f0 on the xz-plane (a,b), yz-plane (e,f), and xy-plane (i,j), for the analytical, (first column) and simulated (second column) lens. Normalized power distribution at fexp on the xz-plane (c), yz-plane (g), and xy-plane (k), for the measured results (third column). Experimentally measured normalized power distribution at z = FLexp along the x-axis (d) and y-axis (h). (l) Experimentally measured normalized power along the z-axis for x = y = 0 mm, at fexp, where FLexp is marked with a dotted vertical line.
Also  Figure 4, it is found that the focus shape is not totally circular. This is due to the depolarization effect, a well-known phenomenon which states that when a linearly polarized incident beam is focused by a high numerical aperture lens, the electric field in the observation plane will be commonly distorted. [67] It must be also noted that a spatial widening is introduced by the receiver probe since it averages the transversal component of the E-field received on its aperture. Then, a wider focus in the experiment compared to the simulation and analytical results is expected, where an ideal point detector is assumed. Another small difference, which can be related to the different nature of the illumination sources, is found in the DoF, yielding the measured result 57% wider than in simulations. Regarding the FoV, the experimental data reveal no relevant sidebands in the 2.5λ0 to 2.5λ0 xy-frame recorded, so at least its value must be FoV > 5λ0. This is in good agreement with the analytical and simulation studies where no relevant sidebands were found in a 100λ0 to 100λ0 xy-frame. Note that such a large spatial range was not considered in the experimental setup as it would have increased the time needed to record each xy-frame by a factor of 40 at least. This could have introduced unavoidable uncertainties in the experiment due to the thermal drift of the devices.
The focusing properties extracted from analytical, simulation and experimental results are summarized in Table 1.   Table 1 Accepted Article

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Summary of the focus performance. focusing efficiency (yield ≈73%); (ii) reduced side lobes that provide a clean FoV, being the power level of the sidelobes below 5% of the focus power level (SLL = 13 dB); and (iii) focal spot below the diffraction limit, (≈25% below the Rayleigh resolution limit). Accepted Article mm (f).

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Finally, the full study of the lens behavior at the operation frequency is presented in

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Finally, the full study of the lens behavior at the operation frequency is presented in

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This article is protected by copyright. All rights reserved The recorded hotspots presented in Figure 5 (panels c and d) show the sharp contrast between the typical SOL focusing profile, dominated by a prominent sideband, and a focus profile very similar to the diffraction pattern of a conventional diffraction-limited lens.
On the other hand, the focus shown in Figure 5e has a FWHM = 0.28λexp, achieving a remarkable reduction of the focal spot by a factor of almost 2.5 compared to the Rayleigh diffraction limit criterion (= 0.66λexp). Of course, this super-resolution comes at a price: lower power enhancement (= 4.8 dB), lower focusing efficiency (yield ≈ 0.5 %), higher side lobes (SLL = +6 dB, sidelobes higher than the focus) and a limited FoV (FoV = 0.8λexp).
Finally, in Figure 5(f) the diffraction map at z = 12.5 mm is presented and it demonstrates the ability of the metaSOL to develop small hotspots (FWHM = 0.37λexp), surpassing again the diffraction limit (= 0.68λexp) by a factor of 1.8. Contrary to the focus of Figure 5(e), this hotspot is not so narrow, but it achieves a much higher focusing efficiency, being yield ≈ 7.5 %, with a slightly wider FoV (= 1.04λexp). The yield value achieved must be emphasized because nowadays, among the many challenging remaining issues, the most important one is the focusing efficiency.
Moreover, even for the largest-sized SOLs reported, [49] the focusing efficiency is only 5% at most when the hotspot size is below 0.38λ0 ∕NA (in the present case, NA = 0.94).
For the sake of completeness and to allow direct comparison, the focusing properties extracted from the most interesting focal planes at fexp are summarized in Table 2.  This article is protected by copyright. All rights reserved As observed there, a very narrow focus (Focus 2) has small power enhancement and very high SLL with a poor yield, meaning that most of the energy is outside the focus. On the contrary, sacrifycing resolution (Focus 1) one gets an outstanding power enhancement with very small SLL and large yield. Focus 3 is an intermediate case between the other two.

Conclusions
The results obtained verify that a metaSOL in the lower THz range can be successfully designed and fabricated to overcome the classical Rayleigh diffraction limit, allowing a higher resolution than that attainable with conventional THz optics. The proposed metastructure was designed using only two hexagonal unit cells. All the numerical, simulation and experimental measurements are in good agreement demonstrating a full characterization of the focusing performance of the metaSOL.
It has been shown that the fabricated metaSOL has the ability to generate a focal spot at a distance of 8.9λexp away from the metastructure achieving a sharp focus with a transversal resolution of 0.52λexp (≈25% below the resolution limit) a power enhancement of 18.2dB and very low sidelobes (sidelobe of 13dB) while its transmission and focusing efficiency are very high (≈71% and ≈73%, respectively). Moreover, the resolution is below the diffraction limit in a broad fractional bandwidth of 18%. Due to the advantages provided by its ultra-small thickness, lightweightness, high transmission and focusing efficiency, ease of integration and sub-diffraction focal size, the proposed metalens may find application in THz imaging systems where thin, flat, light-load, and high-performance optics with sub-diffraction focusing capabilities is demanded.