Abstract
Coding metasurfaces, composed of only two types of elements arranged according to a binary code, are attracting a steadily increasing interest in many application scenarios. In this study, we apply this concept to attain diffuse scattering at THz frequencies. Building up on previously derived theoretical results, we carry out a suboptimal metasurface design based on a simple, deterministic and computationally inexpensive algorithm that can be applied to arbitrarily large structures. For experimental validation, we fabricate and characterize three prototypes working at 1 THz, which, in accordance with numerical predictions, exhibit significant reductions of the radar crosssection, with reasonably good frequency and angular stability. Besides the radarsignature control, our results may also find potentially interesting applications to diffusive imaging, computational imaging, and (scaled to optical wavelengths) photovoltaics.
Introduction
Metamaterials and metasurfaces^{1,2,3} are artificial materials composed of (3D and 2D, respectively) arrangements of subwavelength inclusions, which are engineered so as to tailor the effective properties in a precise, desired fashion, not necessarily attainable in conventional materials. For instance, by relying on powerful design approaches such as transformation optics^{4}, a desired field manipulation can be engineered via a prescribed local tailoring of the constitutive parameters in the region of interest. On the other hand, abrupt changes over the wavelength scale in the phase, amplitude and/or polarization of a wavefront can be impressed via ultrathin gradient metasurfaces^{5}, thereby extending the conventional Snell’s reflection and refraction laws.
In the above examples, the material properties of the inclusions are primarily constrained by the practical availability and, depending on the application, by power dissipation, whereas the shape of the inclusions is typically retrieved via suitably constrained inversedesign procedures, and can vary across a large parameter space. Recently, the idea of “digitizing” the parameter space of the inclusions, i.e., relying on a limited number of inclusion types, was put forward by Della Giovanpaola and Engheta^{6} and by Cui et al^{7}. for metamaterials and metasurfaces, respectively. Within this framework, particularly intriguing appear the socalled “coding metasurfaces”, in which a binary code is associated with each possible inclusion (unit cell). Over the past few years, these structures have been the subject of intense investigation (see, e.g., refs^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}), and their applications have been recently extended also to the acoustic domain^{26,27}. The reader is also referred to refs^{28,29} for recent comprehensive reviews on the subject, which also explore the fascinating concepts of “programmable” and “informationbased” metasurfaces.
Of particular interest for the present study are the applications to diffuse scattering at THz frequencies^{9,10,11,12,14,15,22,23}. Compared with microwave frequencies, the study of metasurfaces in the THz region might be of benefit for a number of different applications, including imaging, radar, and sensing, in view of the inherent higher spatial resolution, stronger secrecy, penetration capability, interference immunity, and field enhancement. In particular, as THz imaging radars^{30,31} are becoming increasingly common in security and safety applications, the design of lowscattering fixtures is gaining a growing attention. Diffusescattering metasurfaces appear to be an attractive solution in view of their potential conformability (by relying on flexible substrates) as well as the negligible impact on the thermal signature (as opposed, e.g., to absorbers). Besides the reduction/control of the radar signature, they may also find important potential applications to diffusive imaging^{32}, computational imaging^{13,33}, and (at optical wavelengths) to light trapping in photovoltaics^{34}.
The design of a coding metasurface for diffuse scattering represents a fairly complex combinatorial problem, typically addressed via computationally intensive optimization algorithms^{11,20,35}, which do not guarantee finding the global optimum, and may become unaffordable for electrically large structures. In a recent study^{25}, we proposed a suboptimal, computationally inexpensive design procedure based on a class of flat polynomials^{36,37,38}. Here, we apply this approach to the design of coding metasurfaces for diffuse scattering at THz frequencies, and we experimentally validate it via timedomainspectroscopy measurements on fabricated prototypes operating at 1 THz.
Results
The basic idea is schematized in Fig. 1. We consider a metasurface composed of a metalbacked dielectric film patterned with two types of unit cells made of metallicpatch elements (labeled as “1” and “0”) arranged according to a 2D binary coding. We assume that the metasurface is placed in the xy plane of a Cartesian reference system (see Fig. 1) and, unless otherwise specified, that a plane wave is normally impinging along the (negative) zaxis, with ypolarized electric field. Our aim is to engineer the unit cells and the coding in order to attain diffuse scattering, i.e., to scatter the impinging plane wave in all possible directions, as uniformly as possible, so as to avoid the emergence of strong Braggtype peaks.
As previously mentioned, this problem is typically addressed via bruteforce numerical optimization^{11,20,35}. Instead, here we follow a different, recently proposed approach^{25} that directly exploits the spectral properties of certain aperiodic binary sequences known as GolayRudinShapiro (GRS) sequences^{36,37,38}. These sequences are characterized by an absolutely continuous spectral response (which is representative of the desired diffusescattering response), and a socalled “trivial” Bragg spectrum that can be suppressed by suitably tailoring the scattering response of the two basic unitcells^{39}. In particular, if multiple scattering is neglected, such suppression is attained by choosing the two unitcell scattering responses equal in magnitude and 180° out of phase^{39}.
As shown in Fig. 2a, in the chosen configuration the above outofphase condition can be engineered at a desired frequency (1 THz, in our case) by suitably choosing the patch size of the two elements.
The binary coding is generated via a simple deterministic algorithm in two steps. First, an auxiliary binary sequence is generated with the alphabet {−1, 1},
and is mapped onto the {0, 1} alphabet as follows:
The 2D coding is finally obtained via dyadic product of the 1D sequence {α_{n}} by itself. In this study, we consider a sequence length N = 2^{ν}(with ν = 1, 2, ...), for which the above coding is strictly related to the socalled GRS polynomials^{36,37,38}, which are known to exhibit some spectral flatness properties that are especially desirable for our diffusescattering application (see also the discussion in ref.^{25}). More specifically, there are two types of GRS polynomials (henceforth referred to as Ptype and Qtype), which can be explicitly defined via two intertwined recursive relationships^{36,37,38}
initialized by P_{0} = Q_{0} = 1, and with ξ denoting a complexvalued (generally unimodular) variable. It can be shown^{36,37,38}, that the coefficients of the GRS polynomials in Eq. (3) are ±1. In particular, the coefficients of the P_{ν}type GRS polynomials are given by the coding sequence {χ_{n}} in Eq. (1), whereas those pertaining to the Q_{ν}type GRS polynomials are obtained by flipping the bit elements in the second half of the sequence {χ_{n}}^{36,37,38}. In ref.^{25}, we demonstrated that these designs are suboptimal, in the sense that the attainable radarcrosssection (RCS) reduction (by comparison with an unpatterned metallic target of same size) is only few dB distant from that obtained via computationallyintensive numerical optimization^{11,20,35} as well as from theoretically derived tight bounds. In what follows, we provide further numerical evidence of this suboptimal character.
Based on the above algorithm, we design and fabricate three metasurface prototypes, with the coding chosen according to the P_{2}, P_{5}, and Q_{5} GRS polynomials. The coding sequences and corresponding patterns are explicitly given in the Supplementary Information (Table S1, Fig. S1). Specifically, the GRS P_{5} and Q_{5}type codings consist of 32 × 32 bit elements, whereas the P_{2}type coding consists only of 4 × 4 elements, and is replicated 8 × 8 times in order to cover the same area as the other two. In view of this longrange periodicity, the P_{2}type coding metasurface is not particularly suited for diffuse scattering but, as it will be clear hereafter, it is mainly considered for calibration purposes.
Figure 2b displays a detailed view of an optical microscope image of a fabricated sample. Note that each bit element is actually a “supercell” made of several (6 × 6, in our case) elementary unit cells, so as to establish a local periodicity on the scale of a wavelength. This is necessary in order to ensure the selfconsistency of the model, since the physical unit cells are designed via fullwave simulations assuming infinite periodic structures (see ref.^{25} for more details).
For the experimental characterization, we utilize a timedomain spectroscopy system. More specifically, with the experimental setups detailed in the Supplementary Information (see the schematics in Fig. S2 and renderings in Fig. S3), we measure the scattered field intensity I_{MS} from the metasurface as a function of the frequency f and direction θ (in the ϕ = 0 plane parallel to the patch side; cf. Fig. 1). As a reference, we also measure the intensity I_{metal} reflected from an unpatterned metal region of same size, and calculate the RCS ratio
with θ_{s} denoting the specularreflection direction (at which the reflectedintensity from the unpatterned metal reference is maximum).
Figure 3 shows the results for normalincidence and backscatteringdirection (θ = θ_{i} = 0, with electric field parallel to the patch side, obtained with the setup in Fig. S2a in the Supplementary Information), as a function of frequency, for the P_{2} and P_{5}type prototypes. Note that the metasurface size is 9.6 × 9.6 mm^{2} (i.e., ~32 wavelengths per linear dimension at the design frequency of 1 THz), and a fullwave simulation of the entire structures is unaffordable with our current computational resources. Accordingly, we utilize instead the semianalytical model detailed in ref.^{25} (see also the Methods section below for more details). Such model, based on a physicaloptics approximation, has been shown to provide a good agreement with fullwave simulations and measurements^{25}. Also in the present case, as it can be observed from Fig. 3, the agreement between measurements and simulations is fairly good, and indicates a significant reduction of the RCS (~10 dB and ~20 dB, for the P_{2} and P_{5}type, respectively), over a sizable frequency region around 1 THz. The deterioration of the agreement at higher frequencies is attributable to the various approximations and unmodeled effects in the simulations, as well as uncertainties in the parameters and fabrication tolerances. The rather strong reflection peak around 1.1 THz (especially visible in the response of the P_{5}type sample in Fig. 3b) is attributable to watervapor absorption (see also the discussion below).
The anticipated poorer performance of the P_{2}type design can be understood from the results in Fig. 4. More specifically, the falsecolorscale maps in Fig. 4a and b show the simulated and measured (via the setup in Fig. S2b in the Supplementary Information), respectively, RCS ratio for normal incidence, as a function of the observation angle and frequency, whereas Fig. 4c and show two representative frequency and angular cuts, respectively. Once again, a generally good agreement between simulations and measurements is observed. In view of the replicationinduced longrange periodicity, the P_{2}type coding metasurface exhibits a Braggtype spectrum with rather sharp peaks, which obviously implies quite poor performance as a diffusive scatterer. In fact, this design mainly serves for calibration purposes, in order to demonstrate that our measurement setup is actually capable of capturing sharp Braggtype peaks whenever present. As it can be observed from Fig. 4d, both Bragg peaks appearing in the accessible region are captured by our measurements. Also shown is the backscattering measurement sample (extracted from Fig. 3a), which falls within the blind region inaccessible by the detector (see Fig. S2b in the Supplementary Information), and is once again in good agreement with the numerical prediction. Note that, in the measured map (Fig. 4b), the distinctive angleindependent scattering features around the frequencies of 1.1, 1.15, and 1.4 THz correspond to wellknown watervapor absorption peaks^{40}. Moreover, the presence of a modal branch that does not appear in the simulated map (Fig. 4a) is likely attributable to the imperfect suppression of a trivial Braggmode^{39}.
Figure 5 shows the corresponding results for the P_{5}type design. In this case, the structure does not exhibit longrange order, and this yields a rather flat response, devoid of Braggpeaks, in line with the desired diffusescattering behavior.
Although the metasurfaces are designed to work at normal incidence, it is interesting to study their sensitivity to obliqueincidence conditions. Figure 6 illustrates, for the P_{2} and P_{5}type designs, the results for oblique incidence of 10° and 20° (while maintaining the electric field parallel to the patch side), at the design frequency of 1 THz. Once again, measurements and simulations are in reasonably good agreement, and the Braggpeaks appearing in the P_{2}type case (Fig. 6a and b) are accurately captured in position and magnitude. Once again, the broader character of the measured peaks (by comparison with the simulated ones) can be attributed to the various approximations and unmodeled effects in the numerical simulations. Prominent among them are the finitesize of the impinging beam (assumed as a plane wave in the simulations) and the elementcoupling (multiple scattering) effects. In connection with the P_{5}type design (Fig. 6c and d), we remark the absence of strong spectral features, which is indicative of a rather good angular stability.
Results qualitatively similar to the P_{5}type design are also observed for the Q_{5}type case, as shown in Figs S4–S6 in the Supplementary Information.
In order to quantitatively assess the optimality of the proposed design, a meaningful observable is the (worstcase) RCS ratio for normal incidence
which represents the maximum scattering intensity (along all possible directions) normalized with respect to that backscattered by an unpatterned metallic surface of same size. Clearly, the smaller this value, the more effective the coding metasurface in attaining diffuse scattering.
By assuming a metasurface composed of N × N square supercells of sidelength d, and applying the semianalytical modeling detailed in ref.^{25}, in view of the wellknown spectralflatness properties^{36,37,38} of the GRS polynomials, it can be shown^{25} that the maximum RCS ratio in Eq. (5) approximately scales as
where C is a constant essentially dependent on the supercell size, and λ = c/f is the vacuum wavelength (with c denoting the corresponding wavespeed).
As previously mentioned, a bruteforce numerical optimization of the coding pattern, so as to minimize the RCS ratio in Eq. (5), is only possible for moderately sized structures, but may become computationally unaffordable for electrically large structures. Moreover, given the inherently nonquadratic character of the optimization problem, numerical algorithms tend to be prone to false solutions, and thus there is no guarantee to attain a global minimum. In ref.^{7}, a hybrid numerical optimization approach was exploited for structures of (linear) electrical size up to ~20λ. In ref.^{25}, via a numerical fit, we showed that the scaling law of the RCS ratio pertaining to these optimized structures was qualitatively similar to that in Eq. (6). Although the numerical fit was based on electrical sizes smaller than those of interest in the present study, its extrapolation can still be assumed as a meaningful benchmark. Accordingly, in what follows, we consider such empirical scaling law
and refer to that as “numerical optimization extrapolation” (NOE).
For the geometry and parameters as in Fig. 2, Fig. 7 shows the numerically computed RCSratio scaling laws pertaining to the GRS Ptype and Qtype designs, with orders ν = 5, 6, 7, 8 (i.e., linear electrical size ranging from 32λ to 256λ), compared with the NOE prediction in Eq. (7). For both P and Qtype GRS coding designs, the loglog scale of the plot highlights an algebraic decay in line with the theoretical predictions in Eqs (6) and (7), with only slight differences on the order of ~5 dB.
We can therefore conclude that our proposed GRSbased coding design is suboptimal, in the sense that it provides a RCS reduction that is only slightly worse than the NOE prediction, irrespective of the electrical size. However, it is important to stress that, by comparison with bruteforce numerical optimization^{7}, our fully deterministic design approach [essentially relying on Eqs (1) and (2)] requires only a negligible computational effort, and can therefore be applied to arbitrarily large structures.
Discussion
To sum up, we have presented the suboptimal design, fabrication and experimental characterization of coding metasurfaces acting as diffusive scatterers at THz frequencies. Our results extend the experimental validation of our general design approach to the THz band. It is worth pointing out that, while our previous experimental validation at microwave frequencies^{25} was limited to relatively small electrical sizes (about 8 wavelengths per linear dimension) due to inherent limitations of our measurement setup, the THz structures characterized here are considerably larger (32 wavelengths), thereby providing a stronger validation of our proposed design. This is crucially important, as our design approach becomes computationally attractive especially in the limit of electrically large structures.
Overall, our THz study confirms the possibility to effectively design electrically large diffusive scatterers via a simple, deterministic and computationally cheap algorithm, with performance comparable with that attainable via computationally expensive bruteforce optimization. Moreover, a reasonably good frequency and angular stability is observed, with ample room for improvement via suitable optimization of the basic unitcells. Current and future research is aimed at exploring more flexible coding strategies (e.g., multibit) for wideband and wideangle optimization, as well as possible applications to diffuse imaging and computational imaging. Also of great interest is the extension of these results to optical wavelengths, with possible applications to light trapping in photovoltaics.
Methods
Numerical Modeling
The reflectioncoefficient phase responses of the unitcells (Fig. 2a) are obtained via finiteelement fullwave simulations, by means of the commercial software package Ansys® HFSS (Electromagnetics suite Release 16.2.0, www.ansys.com/Products/Electronics/ANSYSHFSS). In these simulations, an infinite structure is assumed, with master/slave periodicity boundary conditions (or phaseshift walls, for oblique incidence) at the four sides of the unit cell, and an air box of thickness 260 μm with a porttype termination on top of the metal patch. The electric field is assumed as parallel to the patch side (i.e., yoriented, in the reference system of Fig. 1). For the dielectric film, a nondispersive model is utilized, with relative permittivity ε_{r} = 3 and losstangent \(\tan \,\delta =0.01\), whereas the metal is assumed as perfectly electric conducting. The structure is discretized via the default adaptive meshing (with maximum element size of ~630 nm), which results in about 7,000 degrees of freedom.
For the simulation of the entire metasurfaces, the semianalytical model detailed in ref.^{25} is utilized, with the singleelement response given by the fullwave simulations above.
Prototype Fabrication
First, a metallic film (10 nm titanium and 200 nm gold) is deposited onto a silicon wafer via electronbeam evaporation. Then, a 20 μmthick polyimide layer is coated on the gold film, and is baked on a hot plate at 80, 120, 180, and 250 °C for 5 minutes each. Since the liquid polyimide utilized (Yi Dun New Materials Co. Ltd, Suzhou, China) can only be used to form a maximum thickness of 10 μm at the minimum spin rate of 1150 rpm, the above spincoating and curing processes are repeated twice for the final completion of the 20 μmthick polyimide layer. To enable a good formation of the metallic pattern during the final liftoff process, a dualphotoresist approach is adopted, which includes a successive coating of LOR10A and AZ5214 photoresists, each followed by a softbake process on a hotplate. An ultraviolet exposure and develop processes help transfer the mask pattern to the photoresist. Next, another 10/200 nm titanium/gold layer is deposited via electronbeam, followed by an ultrasonic bath in acetone to form the final metallic pattern.
Measurements
The metasurface characterization is carried out by means of a customized fibercoupled THz timedomain spectrometer.
For backscattering measurements (Fig. 3), the setup schematized in Fig. S2a (Supporting Information) is utilized, in which the THz beam normally impinges onto the target after passing through a beamsplitter, and the backscattered signal is collected along the orthogonal direction. For angular scattering measurements (Figs 4–6, S5 and S6), the setup in Fig. S2b (Supporting Information) is utilized, in which the THz beam directly impinges onto the target (normally or obliquely), and the scattered signal is measured along a circular path, with the detector positioned on a goniometer. In all measurements, the electric field is maintained parallel to the patch side (i.e., yoriented in the reference system of Fig. 1).
Due to the physical size of the components, there is a “blind” region of about 20° between the emitter and detector, schematized as a pinkshaded area in Fig. S2b (Supporting Information), which is not accessible to measurements.
For each configuration, an unpatterned metallic reference is also characterized, in order to calculate the RCS ratio in Eq. (4).
A more detailed description of the experimental setups, as well as the data processing, is provided in the Supplementary Information. Here, we limit ourselves to highlight that the use of collimating planar lenses for both the emitter and receiver, and of a simple goniometric guiding trail for the receiver antenna allows to reach an unprecedented angular resolution (1°) for both the impinging and scattered signals, by comparison with previous schemes based mostly on offaxis parabolic mirrors^{9,10,11,12,15}, where the precise determination of the incident and detection angles is more complex.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.
References
Capolino, F. Theory and Phenomena of Metamaterials. (CRC Press, 2009).
Cai, W. & Shalaev, V. M. Optical Metamaterials: Fundamentals and Applications. (Springer, 2010).
Zhang, L., Mei, S., Huang, K. & Qiu, C.W. Advances in full control of electromagnetic waves with metasurfaces. Adv. Opt. Mater. 4, 818–833 (2016).
Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006).
Yu, N. et al. Light propagation with phase discontinuities: Generalized laws of reflection and refraction. Science 334, 333–337 (2011).
Della Giovampaola, C. & Engheta, N. Digital metamaterials. Nat. Mater. 13, 1115–1121 (2014).
Cui, T. J., Qi, M. Q., Wan, X., Zhao, J. & Cheng, Q. Coding metamaterials, digital metamaterials and programmable metamaterials. Light Sci Appl 3, e218 (2014).
Kaina, N., Dupré, M., Lerosey, G. & Fink, M. Shaping complex microwave fields in reverberating media with binary tunable metasurfaces. Sci. Rep. 4, 6693 (2014).
Yan, X. et al. Broadband, wideangle, lowscattering terahertz wave by a flexible 2bit coding metasurface. Opt. Express 23, 29128–29137 (2015).
Gao, L.H. et al. Broadband diffusion of terahertz waves by multibit coding metasurfaces. Light Sci. Appl. 4, e324 (2015).
Dong, D. S. et al. Terahertz broadband lowreflection metasurface by controlling phase distributions. Adv. Opt. Mater. 3, 1405–1410 (2015).
Liu, S. et al. Anisotropic coding metamaterials and their powerful manipulation of differently polarized terahertz waves. Light Sci. Appl. 5, e16076 (2016).
Li, Y. B. et al. Transmissiontype 2bit programmable metasurface for singlesensor and singlefrequency microwave imaging. Sci. Rep. 6, 23731 (2016).
Zhao, J. et al. Controlling the bandwidth of terahertz lowscattering metasurfaces. Adv. Opt. Mater. 4, 1773–1779 (2016).
Zhang, Y. et al. Broadband diffuse terahertz wave scattering by flexible metasurface with randomized phase distribution. Sci. Rep. 6, 26875 (2016).
Cui, T. J., Liu, S. & Li, L. L. Information entropy of coding metasurface. Light Sci. Appl. 5, e16172 (2016).
Shen, Z. et al. Design of transmissiontype coding metasurface and its application of beam forming. Appl. Phys. Lett. 109, 121103 (2016).
Yang, H. et al. A programmable metasurface with dynamic polarization, scattering and focusing control. Sci. Rep. 6, 35692 (2016).
Chen, K. et al. Geometric phase coded metasurface: from polarization dependent directive electromagnetic wave scattering to diffusionlike scattering. Sci. Rep. 6, 35968 (2016).
Li, S. J. et al. Ultrabroadband reflective metamaterial with RCS reduction based on polarization convertor, information entropy theory and genetic optimization algorithm. Sci. Rep. 5, 37409 (2016).
Liu, S. & Cui, T. J. Flexible controls of scattering clouds using coding metasurfaces. Sci. Rep. 6, 37545 (2016).
Liang, L. et al. Broadband and wideangle RCS reduction using a 2bit coding ultrathin metasurface at terahertz frequencies. Sci. Rep. 6, 39252 (2016).
Zhao, J., Cheng, Q., Wang, T. Q., Yuan, W. & Cui, T. J. Fast design of broadband terahertz diffusion metasurfaces. Opt. Express 25, 1050–1061 (2017).
Li, L. et al. Electromagnetic reprogrammable codingmetasurface holograms. Nat. Commun. 8, 197 (2017).
Moccia, M. et al. Coding metasurfaces for diffuse scattering: Scaling laws, bounds, and suboptimal design. Adv. Opt. Mater. 334, 1700455 (2017).
Xie, B. et al. Coding acoustic metasurfaces. Adv. Mater. 29, 1603507 (2017).
Xie, B. et al. Multiband asymmetric transmission of airborne sound by coded metasurfaces. Phys. Rev. Applied 7, 024010 (2017).
Liu, S. & Cui, T. J. Concepts, working principles, and applications of coding and programmable metamaterials. Adv. Opt. Mater. 5, 1700624 (2017).
Cui, T. J., Liu, S. & Zhang, L. Information metamaterials and metasurfaces. J. Mater. Chem. C 5, 3644–3668 (2017).
Cooper, K. B. et al. THz imaging radar for standoff personnel screening. IEEE Trans. Terahertz Sci. Technol. 1, 169–182 (2011).
Zhang, B., Pi, Y. & Li, J. Terahertz imaging radar with inverse aperture synthesis techniques: system structure, signal processing, and experiment results. IEEE Sensors J. 15, 290–299 (2015).
Jesacher, A., Roider, C., RitschMarte, M., Bernet, S. & Harm, W. Lensless imaging through thin diffusive media. Opt. Express 22, 22146–22156 (2014).
Hunt, J. et al. Metamaterial apertures for computational imaging. Science 339, 310–313 (2013).
Green, M. A. Lambertian light trapping in textured solar cells and light‐emitting diodes: analytical solutions. Prog. Photovoltaics Res. Appl. 10, 235–241 (2002).
Zhao, Y. et al. Broadband diffusion metasurface based on a single anisotropic element and optimized by the simulated annealing algorithm. Sci. Rep. 6, 23896 (2016).
Golay, M. J. E. Static multislit spectrometry and its application to the panoramic display of infrared spectra. J. Opt. Soc. Am. 41, 468–472 (1951).
Shapiro, H. S. Extremal problems for polynomials and power series. Master’s Thesis (Massachusetts Institute of Technology, 1952).
Rudin, W. Some Theorems on Fourier Coefficients. Proc. Am. Math. Soc. 10, 855–859 (1959).
Galdi, V., Castaldi, G., Pierro, V., Pinto, I. M. & Felsen, L. B. Scattering properties of onedimensional aperiodicallyordered strip arrays based on twosymbol substitutional sequences. IEEE Trans. Antennas Propagat. 55, 1554–1563 (2007).
Fattinger, C., Grischkowsky, D. & van Exter, M. Terahertz timedomain spectroscopy of water vapor. Opt. Lett. 14, 1128–1130 (1989).
Author information
Authors and Affiliations
Contributions
T.J.C., V.G. and A.A. conceived the idea, analyzed the results, and supervised the study. M.M. performed the design and numerical simulations, with the assistance of G.C. and inputs from all authors. S.L., L.Z. and R.Y.W. fabricated the prototypes. C.K. conducted the experiments, with the assistance of G.P.P. and A.A. V.G. wrote the manuscript, with inputs from all authors. All authors reviewed the manuscript.
Corresponding authors
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Moccia, M., Koral, C., Papari, G.P. et al. Suboptimal Coding Metasurfaces for Terahertz Diffuse Scattering. Sci Rep 8, 11908 (2018). https://doi.org/10.1038/s4159801830375z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4159801830375z
Further reading

Metasurfaces for Stealth Applications: A Comprehensive Review
Journal of Electronic Materials (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.