Open Access
Add to BibSonomy
Abstract
Lenses with a tunable focus are highly desirable but remain a challenge. Here, we demonstrate a microwave varifocal meta-lens based on the Alvarez lens principle, consisting of two mechanically movable tri-layer metasurface phase plates with reversed cubic spatial profiles. The manufactured multilayer Alvarez meta-lens enables microwave beam collimation/focusing at frequencies centered at 7.5 GHz, and shows one octave focal length tunability when transversely translating the phase plates by 8 cm. The measurements reveal a gain enhancement up to 15 dB, 3-dB beam width down to $3.5^\circ$, and relatively broad 3-dB bandwidth of 3 GHz. These advantageous characteristics, along with its simplicity, compactness, and lightweightness, make the demonstrated flat Alvarez meta-lens suitable for deployment in many microwave systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
During the past decade metasurfaces have demonstrated extraordinary capability of controlling the amplitude, phase, and polarization of electromagnetic waves [1]. In particular, the phase of scattered waves can be tuned to cover a $360^\circ$ range by tailoring the geometry and orientation of subwavelength resonant structures (i.e., meta-atoms). This paradigm provides an unprecedented opportunity to miniaturize optical devices and elements, significantly reducing volume and weight while increasing operational bandwidth [2]. In earlier proof of concept demonstrations, single-layer nanoplasmonic structures were used with anisotropic resonant responses [3], leading to low efficiencies in cross-polarization conversion that is responsible for the required phase delays. This issue has been circumvented in the optical regime by utilizing single-layer all-dielectric metasurfaces where each individual dielectric resonator functions as a micro-half-waveplate [4] and may also take advantage of Pancharatnam-Berry (P-B) phase [5,6]. In the mid-/far- infrared and microwave ranges, the solution can be few-layer metal-dielectric-metal metasurfaces forming a Fabry-Pérot cavity [7,8]. In the latter, the resonator array was either backed with a metal reflector or sandwiched between two orthogonal metal grating polarizers, resulting in dramatically improved efficiency and bandwidth when serving as a wave plate [8,9] or meta-lens [10,11].
Typical meta-lenses suffer from chromatic dispersion, i.e., the focal length varies with the operational wavelength. While direct far-field imaging is not typical at microwaves, a varifocal meta-lens can improve microwave collimation (i.e., beam directivity) and focusing for scan imaging, detection and ranging, as well as more efficiently transmitting/receiving information in communications, particularly when multiple frequencies are utilized. Previous works have shown mechanically reconfigurable focal length when metasurfaces were fabricated on or embedded in a stretchable substrate [12–14]. A zoom meta-lens can be realized by mechanically adjusting the separation between two meta-lens elements along the optical axis, e.g., via a microelectromechanical system (MEMS) platform [15]. Such an approach may be constrained by the limited longitudinal translation distance at microwave frequencies. Varactor diodes have been widely utilized to provide an electrically tunable phase delay [16,17], although this method is mostly suitable for microwave reflectarrays and requires complex metasurface structures and bias circuits [18]. Thus, we need to seek alternative paths to realize low-profile transmissive varifocal meta-lenses for microwave applications.
This challenge is reminiscent of a varifocal scheme proposed independently by Alvarez in 1967 and Lohmann in 1964 via transverse mechanical translation [19,20], where the lens consists of two tandem optical phase plates whose cubic thickness profile is reversed (i.e., complementary) with respect to the other. The tunable focal length is provided by their mutual lateral displacement, and thus Alvarez lenses are naturally broadband but not achromatic. At microwave frequencies, the lateral translation can overcome the space constrains in the longitudinal direction, but the complex thickness profiles impose significant volume and weight as well as manufacturing cost. In this regard, an Alvarez lens based on planar and ultrathin metasurfaces can be advantageous by addressing many of the aforementioned issues. There have been only a few experimental works on Alvarez meta-lens in the optical regime [21,22], where MEMS platforms can be implemented to ease the operation [23,24]. However, the all-dielectric metasurfaces utilized are unsuitable at microwave frequencies due to fabrication complexity and costs compared to metallic metasurfaces based on printed circuit board (PCB) technology. Recently Guo et al. designed a metallic meta-lens utilizing P-B phase at microwave frequencies, which demonstrated tunable focus by rotating two metasurface plates with respect to each other [25]; however, the design works only for circularly polarized waves. Here, we design and experimentally demonstrate a varifocal Alvarez meta-lens based on PCB technology for commonly utilized linearly polarized microwaves at C-band. Using a fabricated prototype Alvarez meta-lens, we show over one octave tuning range of the focal length by mechanically translating the metasurface phase plates, achieving a gain enhancement up to 15 dB, 3-dB beam width down to $3.5^\circ$, and relatively broad 3-dB bandwidth of $\sim$3 GHz.
2. Design of Alvarez meta-lens
We start by taking two-dimensional cubic phase profiles [19,26] for the two cascaded phase plates:
where $(x,y)$ are the transverse plane coordinates within the phase plates, $\phi _{\mathrm {o},1}$ and $\phi _{\mathrm {o},2}$ are the phase values at the origin $O$ which can be set to zero, and $A$ is a positive constant that determines the rate of lens power variation when laterally displacing the phase plates. Under the thin lens approximation, the phase delay of the Alvarez lens is the sum of the two plates. When the two plates are completely aligned, the phase delay is constant ($\phi (x,y) = 0$), indicating an infinite focal length. When the two phase plates are translated by a distance $d$ in opposite directions along the $x$-axis, the phase profiles becomeThe spatial profile of the overall phase delay is then given by adding Eqs. (3) and (4):
where $\rho ^2=x^2+y^2$. The first term in the right side of Eq. 5 is position independent and does not contribute to focusing; the second quadratic term represents a spherical lens that provides the phase advance to compensate the relative phase delay between waves propagating from the lens at positions $(x,y)$ and $(0,0)$ to the focal point, where $\omega$ is the angular frequency and $c$ is the speed of light in free space. The focal length $Z_F$ is then given byFor small values of $Ad$, the focal length is mainly determined by the first term in the right side of Eq. 7 and varies as $Z_F \propto 1/d$, with deviation given by the second term similar to spherical aberration. We design the Alvarez meta-lens to operate at $\omega = 2 \pi \times 7.5$ GHz and take $A = 7.85 \times 10^{-4}$ rad/cm$^3$, which gives a focal length $Z_F \approx 50$ cm when the metasurface phase plates are translated by $d = 10$ cm. With a meta-lens size $\rho _{\mathrm {max}} \approx 20$ cm, the upper limit of focal length deviation should be less than 4%.
Figure 1(a) illustrates the design schematic of our Alvarez meta-lens, where the incident microwaves are linearly polarized along the vertical direction (i.e., $y$-axis). Each of the two metasurface phase plates is based on a tri-layer metasurface design [8], with the building block (unit cell) shown in Fig. 1(b) and consisting of an array of anisotropic subwavelength metallic resonators sandwiched by a pair of orthogonal metal gratings. For a uniform resonator array, the tri-layer metasurface plate has been experimentally demonstrated to exhibit broadband cross-polarization conversion with near-unity efficiency, and the phase delay is determined by the geometry of the anisotropic resonator unit cell, covering the entire range of $360^\circ$ [8]. An array of spatially varying resonators can thereby create an arbitrary phase spatial profile along the transverse directions, e.g., for highly efficient flat metasurface lenses [10,11]. Here this approach provides a viable route to design the building blocks for creating Alvarez metasurface phase plates with cubic phase profiles. The output of the first Alvarez phase plate serves as the input to the second Alvarez metasurface phase plate, and eventually the linear polarization is maintained as polarization conversion occurs twice. This arrangement enables the required quadratic spatial distribution of the overall phase delay, and can be tuned by transversely translating the two metasurface plates. In order words, this achieves the varifocal Alvarez metasurface lens. Note that in the present design the middle panel contains gratings at both front and back surfaces, thus belonging to phase plates 1 and 2, respectively.
Fig. 1. (a) Schematic of a multilayer metasurface Alvarez lens (drawn not to scale) with two movable resonator array panels. The varifocal functionality is realized by mechanically translating the resonator array panels in the transverse (i.e., $\pm x$) directions. (b) Unit cell used in numerical simulations for building block resonators of the tri-layer metasurface phase plates. Insets: split-ring resonator (left) and L-shaped resonator (right), with $L_c$ indicating the cutting distance of the square loop to form the resonators.
As shown in Fig. 1(b), the incident waves polarized perpendicular to the metallic wires of the front grating can reach and interact with the anisotropic resonator array, thereby producing a cross-polarized field component that propagates in both forward and backward directions. There are remaining co-polarized waves in the forward and backward directions as well. For the forward-propagating waves, the rear grating allows the cross-polarized waves to pass, but blocks and reflects the co-polarized waves; for the backward-propagating waves, on the other hand, the front grating allows the co-polarized waves to escape, but blocks and reflects the cross-polarized waves to the forward direction. The tri-layer metasurface can be thus considered as a special Fabry-Pérot cavity [8,27,28], where partial polarization conversion occurs during each of the multiple reflections. The orthogonal configuration of the two gratings results in overall reflection of only co-polarized waves and transmission of only cross-polarized waves. Consequently, designing the building blocks of the metasurface involves optimization of structural dimensions to minimize co-polarized reflection and maximize cross-polarized transmission, and obtain the desired phase delays. This can be achieved by sweeping the structural parameters in numerical simulations or through inverse design using machine learning [29].
Full-wave numerical simulations (CST Studio Suite) were carried out to obtain the cross-polarized transmission amplitude and phase spectra centered at 7.5 GHz, using the configuration shown in Fig. 1(b) with unit cell boundary conditions in the transverse directions and floquet open ports in the wave propagation directions. The metallic structures are made of 17 $\mu$m thick lossy copper on a 0.813 mm thick Rogers RO4003C substrate with a dielectric constant $\epsilon = 3.55$ and loss $\tan \delta = 0.0027$. Through trial and error we determine an anisotropic split-ring resonator structure (SRR; left inset) as the metasurface building block, with unit cell size $P = 10$ mm, grating line width 0.5 mm and period 1 mm, air spacing $s = 5$ mm, and resonator side length $L = 8$ mm and line width $W = 0.5$ mm. Different phase delay is then accomplished through sweeping the size ($L_c$) of the split gap located at the diagonal corner. Increasing $L_c$ to 7.5 mm transforms the SRR to an L-shaped resonator; further increasing $L_c$ reduces the length of the side arms of the L-shaped resonator, as shown in the right inset to Fig. 1(b). Figure 2(a) shows the cross-polarized transmission phase when the sweeping parameter $L_c$ increases from 1 to 8 mm. The phase down-shifts but the curves remain largely parallel with each other over the frequency range between 6 and 9 GHz. The metasurface plate also exhibits a high transmission amplitude over this broad frequency range as revealed in Fig. 2(b). The transmission amplitude and phase at 7.5 GHz are summarized in Fig. 2(c) as a function of $L_c$. The amplitude is higher than $95{\%}$ and the phase is tuned from $294.8^\circ$ to $114.8^\circ$, covering a range of $180^\circ$. An additional $180^\circ$ phase results from flipping the resonator [3,8], thus accomplishing coverage of the entire $360^\circ$ range.
Fig. 2. (a) Phase and (b) amplitude spectra of the cross-polarized transmission through the tri-layer metasurface structure when varying the anisotropic resonator by increasing $L_c$ from 1 mm to 8 mm with a step of 1 mm. (c) Phase (red left arrow indicating left axis) and amplitude (blue right arrow indicating right axis) at 7.5 GHz as a function of $L_c$, along with a second degree polynomial fit of the phase data. Solid and dashed curves in (a,b), as well as filled and unfilled symbols in (c), are for SRRs and L-shaped resonators, respectively.
A standard 24 by 18 inches PCB substrate can fit an array of $60 \times 45$ resonators. The center position of each individual resonator is represented by an integer-index pair $(m,n)$, i.e., $x^{(m)} = (m - 1/2)$ cm and $y^{(n)} = n$ cm, with $m = -29, \ldots, 30$ and $n = -22, \ldots, +22$. The corresponding phase $\phi (m,n)$ is calculated using Eq. 3 with $d = d_0 = 10$ cm, resulting in the spatial profile shown in Fig. 3(a). Here we have purposely shifted the phase profile from the panel center in the $x$-direction by $d_0 = 10$ cm, which serves as an “initial displacement” so that the Alvarez meta-lens has a focal length $Z_{F0} \approx 50$ cm at 7.5 GHz before any mechanical translation. The required phase at each discrete position is then satisfied by choosing a resonator with appropriate $L_c$, as displayed in Fig. 3(b), by solving the second degree polynomial fitting of the simulated phase data in Fig. 2(c): $\phi = C_2 L_c^2 + C_1 L_c + C_0$ where $C_2 = -0.07$, $C_1 = -25.04$, and $C_0 = 320.7$. Note that $C_0$ can be freely altered to offset the phase curve so that any resonator could serve as phase zero. Also note that we have ignored here the effect on the scattering phase induced by the couplings to the spatially varying neighbors. Identical resonator array panels are used for both metasurface phase plates 1 and 2, by simply rotating one of the panels by $180^\circ$ along its $y$-axis, thus transforming the coordinate $x \rightarrow -x$ and consequently Eq. 3 to Eq. 4.
Fig. 3. (a) Calculated cubic phase spatial profile for designing the metasurface phase plate. (b) A photo of the fabricated resonator array panel with spatially varying resonators in accordance with the design. Identical resonator array panels are used for both phase plates 1 and 2, as explained in the main text.
3. Experiments and results
The Alvarez lens phase plates were manufactured using standard photolithography-based PCB fabrication technologies in accordance with the design parameters except for the substrate thickness of 1.524 mm for the middle double-sided grating panel. A photograph of the fabricated resonator array panels is shown in Fig. 3(b). These panels were then assembled with 5 mm air spacing to form the Alvarez meta-lens, which was characterized inside an anechoic chamber using a pair of broadband horn antennas (SAS-571) and a vector network analyzer (Agilent N5230A), as schematically shown in Fig. 4(a) for the experimental setup. The Alvarez meta-lens and the transmitting horn antenna (Tx) were separated by a distance $Z$ along the optical axis and placed on a computer-controlled gimbal, which can rotate in both horizontal (azimuth or pan) and vertical (elevation or tilt) directions as indicated by the red and blue arrows, respectively. The receiving horn antenna (Rx) was fixed and located at $\sim$11 m away. Through the manual change of the distance $Z$ and the computer-controlled scanning of the pan and tilt angles, we retrieved the relative gain and radiation patterns of the Alvarez meta-lens, in which the reference was obtained with the Alvarez meta-lens removed.
Fig. 4. (a) Schematic of the Alvarez meta-lens characterization system in an anechoic chamber. (b) Measured relative gain at 7.5 GHz as a function of $Z$ at different values of metasurface phase plate displacement $d$. (c) The same as in (b) but measured at 6.0 GHz.
At the designed frequency of 7.5 GHz, the boresight gain as a function of lens-Tx distance $Z$ is shown in Fig. 4(b) with different values of displacement $d$. When the two metasurface phase plates are completely overlapped with each other (i.e., $d = d_0 = 10$ cm), the peak gain of 11 dB occurs at $Z_F = 49$ cm, which is close to the designed focal length of 50 cm. For $d = 6$ and 14 cm (i.e., translating the resonator array panels $\mp 4$ and $\pm 4$ cm), the retrieved focal lengths are approximately 56 and 34 cm, respectively, with gains about 10 dB. Compared to the designed focal lengths of 83 and 36 cm, the deviation is significant particularly for $d = 6$ cm, which is in part due to the less than ideal gain peaks, making it difficult to accurately determine the focal lengths.
When assembling the Alvarez meta-lens for measurements, we accidentally made the outer gratings face backward rather than toward the resonator array panels as in the design we illustrate in Fig. 1. This results in an additional spacing between the metallic structures, and is expected to slightly down-shift the optimal operational frequency as compared to the designed 7.5 GHz. In Fig. 4(c) we show the gain curves at 6.0 GHz, revealing higher and better defined gain peaks. For $d = 6$, 10, and 14 cm, the peak gains are 11.5, 13.5, and 12.5 dB, and the focal lengths are 52, 35, and 27 cm, respectively. These focal lengths are in good agreement with the expected focal lengths of 67, 40, and 29 cm at 6.0 GHz following the first term in the right side of Eq. (7).
Figure 5(a) and 5(b) show the radiation pattern as a function of pan and tilt angles measured at 7.5 GHz with $d =10$ cm and $Z = 49$ cm. The results reveal about 15 dB boresight gain as compared to the horn antenna alone. The measured higher and more accurate gain also suggests that the alignment in our measurements for Fig. 4 should have been improved. The pan and tilt scans reveal 3-dB beam widths of $3.5^\circ$ and $4.5^\circ$ in the azimuthal and elevation direction, respectively. No significant side lobes (i.e., $< 10$ dB) are observed within the scanning angle ranges, as shown in Fig. 5(c). The two-dimensional plot of the measured beam profile in the figure reveals a highly symmetric and nearly circular beam shape. The Alvarez meta-lens shows 1-dB and 3-dB bandwidths of 1.8 GHz and 2.9 GHz, respectively, which is reasonable considering the frequency-dependent focal length. Additional measurements with configuration for 6.0 GHz exhibit very similar behaviors.
Fig. 5. Radiation pattern after the Alvarez meta-lens measured at 7.5 GHz with $d =10$ cm and $Z = 49$ cm as a function of (a) pan and (b) tilt angles, in comparison to the radiation pattern with horn antenna only. (c) Two-dimensional plot of the measured beam profile after the Alvarez meta-lens.
4. Conclusion
In summary, we have designed and experimentally demonstrated a prototype Alvarez varifocal meta-lens operating at C-band frequencies. The cubic phase spatial profiles of the Alvarez phase plates are based on a tri-layer metasurface design capable of highly-efficient cross-polarization conversion and a full coverage of the $360^\circ$ phase space, which is scalable to operate at terahertz and infrared frequencies. Our experimental results have shown tunable microwave beam focusing/collimation with $\sim$1 octave focal length tunability when moving the metasurface phase plates by 8 cm. The prototype Alvarez meta-lens also exhibits desirable properties including a gain enhancement up to 15 dB, 3-dB beam width down to $3.5^\circ$, and relatively broad 3-dB bandwidth of $\sim$3 GHz. These advantageous characteristics, along with its simplicity, compactness, and lightweightness make the demonstrated flat Alvarez meta-lens well suited for deployment in many microwave systems and have the potential to be extended to terahertz applications.
Funding
Los Alamos National Laboratory (20180062DR, 20220627DI, 20230124ER).
Acknowledgments
We thank Shobitha Kramadhati, Dillon Purcell, and Pulak Nath for their technical assistance. Conrad Corbella Bagot acknowledges partial support by the mobility grants program of Centre de Formació Interdisciplinària Superior (CFIS) - Universitat Politècnica de Catalunya (UPC). This work was performed, in part, at the Center for Integrated Nanotechnologies (CINT), an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is managed by Triad National Security, LLC for the U.S. Department of Energy’s NNSA, under contract 89233218CNA000001.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.
References
1. H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016). [CrossRef]
2. W. T. Chen, A. Y. Zhu, and F. Capasso, “Flat optics with dispersion-engineered metasurfaces,” Nat. Rev. Mater. 5(8), 604–620 (2020). [CrossRef]
3. N. F. Yu, P. Genevet, M. A. Kats, et al., “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
4. S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016). [CrossRef]
5. L. Huang, X. Chen, H. Mühlenbernd, et al., “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12(11), 5750–5755 (2012). [CrossRef]
6. D. M. Lin, P. Y. Fan, E. Hasman, et al., “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014). [CrossRef]
7. S. Sun, Q. He, S. Xiao, et al., “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef]
8. N. K. Grady, J. E. Heyes, D. R. Chowdhury, et al., “Terahertz metamaterials for linear polarization conversion and anomalous refraction,” Science 340(6138), 1304–1307 (2013). [CrossRef]
9. C.-C. Chang, Z. Zhao, D. Li, et al., “Broadband linear-to-circular polarization conversion enabled by birefringent off-resonance reflective metasurfaces,” Phys. Rev. Lett. 123(23), 237401 (2019). [CrossRef]
10. C.-C. Chang, D. Headland, D. Abbott, et al., “Demonstration of a highly efficient terahertz flat lens employing tri-layer metasurfaces,” Opt. Lett. 42(9), 1867 (2017). [CrossRef]
11. A. K. Azad, A. V. Efimov, S. Ghosh, et al., “Ultra-thin metasurface microwave flat lens for broadband applications,” Appl. Phys. Lett. 110(22), 224101 (2017). [CrossRef]
12. H.-S. Ee and R. Agarwal, “Tunable metasurface and flat optical zoom lens on a stretchable substrate,” Nano Lett. 16(4), 2818–2823 (2016). [CrossRef]
13. S. M. Kamali, E. Arbabi, A. Arbabi, et al., “Highly tunable elastic dielectric metasurface lenses,” Laser Photonics Rev. 10(6), 1002–1008 (2016). [CrossRef]
14. A. She, S. Zhang, S. Shian, et al., “Adaptive metalenses with simultaneous electrical control of focal length, astigmatism, and shift,” Sci. Adv. 4(2), eaap9957 (2018). [CrossRef]
15. E. Arbabi, A. Arbabi, S. M. Kamali, et al., “MEMS-tunable dielectric metasurface lens,” Nat. Commun. 9(1), 812 (2018). [CrossRef]
16. H.-X. Xu, S. Ma, W. Luo, et al., “Aberration-free and functionality-switchable meta-lenses based on tunable metasurfaces,” Appl. Phys. Lett. 109(19), 193506 (2016). [CrossRef]
17. A. E. Cardin, S. R. Silva, S. R. Vardeny, et al., “Surface-wave-assisted nonreciprocity in spatio-temporally modulated metasurfaces,” Nat. Commun. 11(1), 1469 (2020). [CrossRef]
18. J. Zhao, X. Yang, J. Y. Dai, et al., “Programmable time-domain digital-coding metasurface for non-linear harmonic manipulation and new wireless communication systems,” Natl. Sci. Rev. 6(2), 231–238 (2019). [CrossRef]
19. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. Patent 3,305,294 (1967).
20. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9(7), 1669 (1970). [CrossRef]
21. A. Zhan, S. Colburn, C. M. Dodson, et al., “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017). [CrossRef]
22. X. Che, Y. Yu, Z. Gao, et al., “A broadband achromatic Alvarez metalens,” Opt. Laser Technol. 159, 108985 (2023). [CrossRef]
23. S. Colburn, A. Zhan, and A. Majumdar, “Varifocal zoom imaging with large area focal length adjustable metalenses,” Optica 5(7), 825 (2018). [CrossRef]
24. Z. Han, S. Colburn, A. Majumdar, et al., “MEMS-actuated metasurface Alvarez lens,” Microsyst. Nanoeng. 6(1), 79 (2020). [CrossRef]
25. Y. Guo, M. Pu, X. Ma, et al., “Experimental demonstration of a continuous varifocal metalens with large zoom range and high imaging resolution,” Appl. Phys. Lett. 115(16), 163103 (2019). [CrossRef]
26. P. J. Smilie, T. J. Suleski, B. Dutterer, et al., “Design and characterization of an infrared Alvarez lens,” Opt. Eng. 51(1), 013006 (2012). [CrossRef]
27. H.-T. Chen, J. F. Zhou, J. F. O’Hara, et al., “Antireflection coating using metamaterials and identification of its mechanism,” Phys. Rev. Lett. 105(7), 073901 (2010). [CrossRef]
28. H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express 20(7), 7165–7172 (2012). [CrossRef]
29. S. Krasikov, A. Tranter, A. Bogdanov, et al., “Intelligent metaphotonics empowered by machine learning,” Opto-Electron. Adv. 5(3), 210147 (2022). [CrossRef]
