# Super-Oscillatory Metalens at Terahertz for Enhanced Focusing with Reduced Side Lobes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}(1.4 times below the resolution limit) at the design focal length of 10λ

_{0}with reduced side lobes (the side lobe level being approximately −11 dB). The metalens is optimized at 0.327 THz, and has been validated with numerical simulations.

## 1. Introduction

_{0}with λ

_{0}as the operation wavelength, considering the Rayleigh criterion—see [1] and Appendix A.2 in this manuscript. To accomplish this, an advanced version of the binary particle swarm optimization (BPSO) algorithm, based on [21], is developed (described in detail in Appendix A.3). The designed SOL is then implemented using only two different unit cells at the operation frequency of f

_{0}= 0.327 THz (λ

_{0}= 917 µm). An increased amplitude of the focal spot and reduced side lobes are obtained with the proposed structure, compared to conventional SOLs based on opaque–transparent zones, without affecting its performance in terms of the subwavelength resolution of the focal spot, with a width of 0.44λ

_{0}and 0.50λ

_{0}along the x and y axis, respectively.

_{0}and 0.25λ

_{0}[22]. Compared with this work, our algorithm is more complex, but can develop solutions with a similar resolution, providing, simultaneously, the actual geometry that fulfills the prescribed field distribution profile. In [23], broadband imaging with a resolution approximately 0.64 times the Rayleigh criterion in the visible range was obtained using metasurface unit cells. However, the phase difference between the zones was fixed at π rad, deterring the transmittance and overall efficiency. It is noteworthy that our lens shows a comparable resolution, but with a higher transmittance efficiency. In [24], a new class of super-oscillation waveform was proposed to obtain lower side lobes, as low as 15 dB below the main spot, at the expense of having a diffraction-limited focus. Our work, in contrast, shows higher side lobes (11 dB below the main beam), but the main spot is not diffraction limited.

## 2. Design and Analytical Results

_{hex}carved on a copper metallic film with conductivity σ = 5.8 × 10

^{7}S/m and thickness h

_{m}= 0.6 µm (0.65 × 10

^{−3}λ

_{0}), laying on a polypropylene slab with relative permittivity ɛ

_{r}= 2.25 and thickness h

_{d}= 35 µm (0.038λ

_{0}). Note that, for simplicity, we used the DC nominal conductivity of copper, although, in the terahertz band, this nominal value is usually lower due to granularity, etc.

_{0}. From these results (shown in Appendix A.4), it is found that a phase difference of at least π/6 rad between even and odd zones is necessary for the BPSO algorithm to converge and generate the focal spot at the design FL. In addition, since our aim is to increase the amplitude at the focus, both unit cells must be carefully engineered to ensure a high transmittance at the operation frequency. To achieve this, both the magnitude and phase of the transmission coefficient must be adjusted by varying the geometrical parameters of the unit cell. For simplicity, here, only the external radius of the hexagonal slot (α) is modified, as we found that tuning this parameter is enough to achieve a satisfactory performance.

^{®}. Unit cell boundary conditions are applied on the transverse plane and open boundaries along z (see coordinate axis in Figure 1a). The structure is illuminated with the fundamental TE

_{00}mode of a Floquet port, which corresponds to a vertically polarized plane wave (E

_{y}), considering only normal incidence. With this setup, the contour maps of the magnitude and phase of the transmission coefficient as a function of α and frequency are calculated and shown in Figure 1b,c, respectively. As observed, the phase of the unit cells can be tuned from −π/2 to π/2 rad within the spectral range under study. Note that the phase excursion does not cover the complete −π to π rad range, as required in a graded index lens design for instance. To increase the phase excursion, one could either increase the frequency range or change the unit cell such that this requirement is fulfilled. However, this is not an issue in the proposed SOL that only requires two unit cells with a small phase difference of π/6 between them. Hence, from these results, we can determine the slot width of each unit cell, taking into account the two design conditions mentioned above, to get the highest possible transmittance and a phase difference larger than π/6 rad at the design frequency of f

_{0}= 0.327 THz (highlighted with a vertical dashed gray line in Figure 1b,c). The selected unit cells correspond to the designs with α

_{1}= 60 µm (0.065λ

_{0}) and α

_{2}= 20 µm (0.022λ

_{0}), which have a relatively high transmission coefficient of 0.8 and a phase of 0.12π and 0.28π, respectively, resulting in a phase difference of 0.16π, fulfilling both requirements. Both solutions have been highlighted with horizontal dashed gray lines in Figure 1b,c.

_{max}= 6, as suggested in [25], to set a limit and prevent further exploration after the population has converged; c

_{1}= c

_{2}= 2, to give equal weight to the social and the cognitive components; and w ∈ [0.4, 0.6], considering a time-varying inertial weight starting from 0.6, and decreasing proportionally after each iteration. The maximum number of iterations is set to 2000. In addition, a swarm of 100 particles is considered, each of them being a vector with 72 components. In order to reduce the computational burden, a cylindrical lens is first designed, and its near-field distribution at the operation frequency is calculated analytically with the Huygens–Fresnel approximation [19,20]. Isotropic point sources, with magnitude and phase taken from the selected unit cells, are placed at each of the 72 positions with a separation of 350 µm (0.381λ

_{0}).

_{0}, starting with a random unit cell distribution [13]. The highest weight in the goal function is given to the focal spot. This makes the variance much bigger there than elsewhere, and the optimizer tries to reduce it, quickly developing a focus at the desired point, as shown in Figure 2b. The second step consists in reducing the power distribution of the side lobes with a new weighted exponential goal function. Another optimization process is launched using particles derived from the best combination found in the first step, and varying, then, one position between two consecutive particles. After applying this two-step process, the found global best solution is 111122222111122211221122212211221221221121121121121121121221221211211212, where “1” and “2” stand for unit cells of type 1 and 2 (α

_{1}and α

_{2}), respectively. This vector is the unit cell distribution along the x-axis, from the center of the lens to its rightmost edge (the left-hand side is obtained by simply mirroring the array, making use of the lens symmetry). Thus, after this procedure, the cylindrical metalens has a total length along the transversal x axis of D ≈ 54λ

_{0}.

_{0}, very near the designed value of FL = 10λ

_{0}. From the power distribution along the x-axis at z = FL, depicted in Figure 3a, we see that the value of full width at half-maximum (FWHM, defined as the distance at which the power distribution has been reduced to half its maximum) in the transversal x direction is FWHM

_{x}= 0.36λ

_{0}, which is well below the diffraction limit (0.65λ

_{0}).

_{0}, and has a FWHM = 0.46λ

_{0}both along x and y directions, also below the diffraction limit.

_{local}. According to the definition of super-oscillation [14], k

_{local}is equal to the phase gradient (k

_{local}= ∇Ψ), where Ψ is computed as Ψ = arg{

**E**(r

_{0})·

**E**(r

_{i})}, the function arg{a} represents the argument of a complex number a, and

**E**(r

_{0})·

**E**(r

_{i}) is the scalar product of the electric field vector at points r

_{0}and r

_{i}; r

_{0}is the reference point with coordinates (0, 0, FL), and r

_{i}has coordinates (x, y, FL). In the super-oscillatory region, k

_{local}should be larger than the highest wavenumber component (k

_{0}= 2π/λ

_{0}, in free space). We use the center of the lens as a reference point to calculate the phase of the electric field along the x-axis at the focus (z = FL). As shown in Figure 4, there are regions where the phase rapidly oscillates, and k

_{local}is much larger than k

_{0}, demonstrating that the operation is based on super-oscillations. Moreover, these regions correspond to the electric field intensity minima, which is also a characteristic of super-oscillatory devices.

## 3. Simulation Results

^{®}is used to evaluate the performance of the full metalens. The center of each unit cell is placed at the coordinates of the source points obtained in the Huygens–Fresnel analysis. A schematic of the final design is shown in Figure 1d,e. The lens is illuminated using a plane wave under normal incidence assuming open boundary conditions in all directions. Moreover, magnetic and electric symmetries are applied on the yz-plane and xz-plane, respectively. A fine hexahedral mesh is used, with smallest and largest mesh cells of 17.8 μm (≈0.019λ

_{0}) and 60 μm (≈0.07λ

_{0}), respectively.

_{0}), which is near the designed value (10λ

_{0}) with a transversal resolution of FWHM

_{x}= 0.44λ

_{0}and FWHM

_{y}= 0.5λ

_{0}along the x- and y-axis, respectively. Note that these values are below the diffraction limit (0.65λ

_{0}). Low side lobes are obtained in all cases, with a magnitude of the highest side lobe approximately 10% below of the main lobe in the simulated SOL. With respect to the depth of focus (DOF, defined as the distance along the z-axis where the power distribution has decayed half its maximum from the FL), the simulated spherical SOL shows a bigger value than the analytical spherical SOL (1.51λ

_{0}and 1.28λ

_{0}, respectively). This is an expected result because of the higher accuracy of the numerical analysis, done using the physical unit cells (taken from Figure 1). To compare, further, the focusing performance, the power enhancement (defined as the power amplitude at the FL with and without the SOL) is smaller in the numerical simulation. This can be explained by considering that, in contrast to the analytical calculation, in the simulated model, both material loss and diffraction effects are considered. Finally, it can be noted that the focus ellipticity (defined as the ratio between FWHM

_{x}and FWHM

_{y}) is very close to unity in both spherical lenses, which means an almost spherical focal spot in the xy plane.

## 4. Conclusions

_{0}has been demonstrated at 0.327 THz. A new algorithm, based on a two-step variation of the BPSO, has been developed to enhance the intensity at the focus, and mitigate the side lobe level. The solution obtained shows super-resolution with reduced side lobes and a large enhancement while allowing a spherical focus, with ellipticity close to unity. Both the analytically and numerically calculated spherical lens create a subwavelength focal spot near the prescribed FL, sharper than 0.5λ

_{0}, beating the Rayleigh diffraction limit (0.65λ

_{0}). Moreover, there are no significant side lobes (around 10% of the central peak power), with a power enhancement at the FL of more than 16 dB.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. The Concept of Super-Oscillation

**Figure A1.**Example of a super-oscillatory function, f(x), in red. Its highest harmonic f

_{highest}(n = 5) is represented in blue. In a narrow interval around x = 0, f(x) is approximated by the function f

_{app}.

#### Appendix A.2. Rayleigh Resolution Limit

_{i}, through the equation tan(θ) = r

_{i}/FL. Considering our design, the Rayleigh criterion for the diffraction limit is res = 0.65λ

_{0}.

#### Appendix A.3. Implemented BPSO Algorithm

_{i}(t), and its velocity is denoted by V

_{i}(t) that describes its speed and direction. Each particle has a memory that saves the personal best position found until then, P

_{i}(t) and the algorithm also saves the best global swarm position, G(t).

_{i}(t) − X

_{i}(t), G(t) − X

_{i}(t), and V

_{i}(t), are calculated for each particle. Then, the particle is moved parallel to those three vectors towards its new updated position, denoted as X

_{i}(t + 1). To do it, a weighted sum of the three vectors is applied to Xi(t) to get the updated position and velocity of the particle, using the next equations:

_{1}and c

_{2}, which are known as the cognitive and the social factor respectively; w known as the inertial weight; and r

_{1}and r

_{2}, which are random uniformly distributed values between 0 and 1. The velocity vector will be a real number between [−V

_{max}, V

_{max}]. Since the particle’s position is a binary N-dimensional vector, it must be updated following Equation (1), restricted to binary (0 or 1) values, following the strategy of [16]. The decision to take 0 or 1 is implemented using the sigmoid limiting transformation.

^{®}. For the purposes of this study, it was developed originally and without dependence on any of the existing libraries, adapting and developing the ideas found in the original PSO algorithm. The program was run in a computer with the following characteristics: one octa-core processor i7-6700k CPU @ 4 GHz, 64 GB of RAM DDR4-2400 MHz, SSD 512 GB and graphic card Gigabyte GeForce GTX 1070 Ti 8 GB GDDR5. The running time using 72 lossless source points (half a cylindrical lens), considering around 4000 iterations (2000 iterations for each one of the two algorithm steps), is approximately 5400 s. Regarding the computational burden, if a quarter of a spherical lens would be considered instead, the number of source points would sum up more than 4400, increasing the computational time to more than 60 times higher.

#### Appendix A.4. The Need of a Minimum Phase Difference between Unit Cells

**Figure A2.**Maps showing the normalized power distribution in the xz-plane considering a phase difference between the even and odd zones unit cells (in rads) of (

**a**) π/90; (

**b**) π/22; (

**c**) π/11; (

**d**) π/7.5; (

**e**) π/6; (

**f**) π/2.5. The transmission coefficient magnitude is set to 0.8.

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**Figure 1.**(

**a**) Hexagonal unit cell proposed along with its geometrical parameters: L

_{hex}= 200 µm, α varying from 20 to 80 µm, Δr = 30 µm, metal thickness h

_{m}= 0.6 µm, dielectric height h

_{d}= 35 µm. The metallic material (in grey) is copper, and the dielectric substrate (in blue) is polypropylene. (

**b**) Normalized magnitude and (

**c**) phase (in radians) maps of the transmission coefficient of the unit cell as a function of the parameter α and frequency. (

**d**) Full metalens schematic and (

**e**) zoomed view of the metalens central zones. In (

**f**,

**g**) are shown the unit cells of the even and odd zones, respectively. (

**h**) Diagram showing the unit cell distribution of the designed cylindrical lens. Green and cyan squares represent a unit cell of type (f) or (g) respectively. For representation purposes, they have been shifted vertically, although, in the designed lens, they are all aligned along the x axis. The dark blue background is included only to enhance the contrast and help visualization.

**Figure 2.**Normalized power distribution after (

**a**) random start; (

**b**) end of 1st step; and (

**c**) end of 2nd step.

**Figure 3.**(

**a**) Numerical results of the normalized power distribution along the x-axis at z = FL = 9.816λ

_{0}for the cylindrical super-oscillatory lens (SOL). (

**b**) Idem for the spherical SOL along both x and y-axes at z = FL = 9.162λ

_{0}.

**Figure 4.**(

**a**) Phase distribution in radians at z = FL = 9.162λ

_{0}, using the central point as phase reference. (

**b**) Local wavenumber-vector distribution (blue) where the peaks indicate the super-oscillatory regions, and electric field intensity (dotted red) along radial direction, normalized to the maximum.

**Figure 5.**Normalized power distribution on the xz-plane (

**a**,

**d**), yz-plane (

**b**,

**e**), and xy-plane (

**c**,

**f**), for both the simulated spherical lens (right column) and the analytical spherical lens (left column). The xy-plane is obtained at z = FL, with FL = 9.56λ

_{0}in the simulated lens, and FL = 9.16λ

_{0}in the analytical lens; the xz-plane is the H-plane, and the yz-plane is the E-plane.

Cylindrical SOL (Analytical) | Spherical SOL (Analytical) | Spherical SOL (Simulation) | |
---|---|---|---|

FL | 10λ_{0} | 9.16λ_{0} | 9.56λ_{0} |

FWHM_{x} | 0.36λ_{0} | 0.46λ_{0} | 0.44λ_{0} |

FWHM_{y} | - | 0.46λ_{0} | 0.5λ_{0} |

Enhancement | - | 18.4 dB | 16.6 dB |

Ellipticity | - | 1 | 0.88 |

Depth of Focus | 1.14λ_{0} | 1.28λ_{0} | 1.51λ_{0} |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Legaria, S.; Pacheco-Peña, V.; Beruete, M.
Super-Oscillatory Metalens at Terahertz for Enhanced Focusing with Reduced Side Lobes. *Photonics* **2018**, *5*, 56.
https://doi.org/10.3390/photonics5040056

**AMA Style**

Legaria S, Pacheco-Peña V, Beruete M.
Super-Oscillatory Metalens at Terahertz for Enhanced Focusing with Reduced Side Lobes. *Photonics*. 2018; 5(4):56.
https://doi.org/10.3390/photonics5040056

**Chicago/Turabian Style**

Legaria, Santiago, Victor Pacheco-Peña, and Miguel Beruete.
2018. "Super-Oscillatory Metalens at Terahertz for Enhanced Focusing with Reduced Side Lobes" *Photonics* 5, no. 4: 56.
https://doi.org/10.3390/photonics5040056