Open Access
Add to BibSonomy
Abstract
Metagrating, a periodic metamaterial structure proposed in recent years, is characterized by its structural simplicity and sparsity, compared to a metasurface. It is able to guarantee very high efficiency, even for wide-angle beam deflections, where only a few meta-atoms are required. In such cases, numerical optimization can be avoided and our goal in this work is to provide a fully analytical design study of a metagrating containing only two meta-atoms in a supercell. A series of closed-form design formulas are given, such as the impedance density expression of the meta-atoms, the passive and lossless conditional equations, as well as equations for the power ratio control of the diffraction modes. Four metagratings composed of microstrip capacitor structures working at 10 GHz for wide-angle beam steering and beam splitting are numerically demonstrated. The simulation results agree well with the theoretical predictions, which validates the correctness and effectiveness of the proposed theoretical method.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wavefront control has always been a focus of interest in the electromagnetic and optical fields. Periodic structures known as diffraction gratings are well known for diffracting incident light [1]. Depending on the angle of incidence and the period length of the grating, it diffracts incident light into several beams that propagate at different angles. Based on these properties, gratings have been proposed for applications in holography [2] and wavefront sensing [3]. Furthermore, three-dimensional structure consisting of sawtooth structures or grooves called echelette or blazed gratings [4,5], have been designed to guide incident waves to a particular diffraction mode. On the other hand, metasurfaces as two-dimensional structures, also present excellent performances in wavefront control [6–9]. Based on the generalized Snell’s law, a phase gradient can be implemented in a metasurface so as to control the incident wavefront and thus change the reflected/transmitted beam direction [10]. However, the design based on this method presents the drawback of wave impedance mismatching between incident and outgoing waves for large-angle beam deflection, which leads to a significant efficiency reduction and thus posing severe challenges for practical applications [11,12].
Recently, metagratings [13,14] have been proposed by combining diffraction gratings with the design concept of meta-atoms in metasurfaces. Metagratings can achieve complete control of the beam in each diffraction channel by judiciously designing the meta-atoms, and can moreover maintain nearly 100% efficiency when the outgoing beam is deflected to a large angle, which greatly overcomes the shortcomings of conventional metasurfaces. In the last recent years, the development of metagrating has known great progress [15–17]. In previous studies, a single-wire metagrating based on a capacitive structure was analytically designed to realize the wide-angle beam deflection under oblique wave incidence [18], equal-power beam splitting [19] and absorption [20,21] under normal wave incidence, and the analog optical computing and subwavelength imaging based on metagrating are presented in the literature [22] and [23], respectively. In order to control more diffraction modes for beam deflection or beam splitting under normal incidence, more meta-atoms were introduced into the period (supercell) of the metagratings [24–27]. Moreover, a metagrating anomalous reflector using dual-resonant meta-atoms that can operate in two different frequency bands was presented in [28], and an electrically tunable transmissive metagrating was proposed in [29]. A detailed semi-analytic methodology for designing arbitrary multilayer and multiple meta-atoms metagrating was also given in [30]. Meanwhile, an equivalent circuit method to model and design a metagrating from an arbitrary number of meta-atoms was proposed in [31].
In general, when designing metagratings, the number of design degrees of freedom depend on the number of diffraction modes that need to be handled. In the microwave frequency band, adding more metallic scatters in the supercell allows to increase the number of degrees of freedom, while dielectric meta-atoms with multipolar characteristics are used in optics. When a large number of degrees of freedom are used, various optimization methods can be useful to implement the desired functionalities. However, for large angle beam deflection or beam splitting to a limited number of beams, especially at microwave frequencies, which can present a high potential application value in antenna or reflector plate designs, a relatively small number of metallic meta-atoms can be sufficient in the metagrating’s supercell. Though an analytical design methodology for beam deflection under obliquely incident illumination has been detailed in [18], still no fully analytical design methodology was presented for beam deflection metagratings operating under normal incidence. In the case of normal incidence, due to the symmetry of the diffraction modes, at least two diffraction modes need to be suppressed in order to achieve the beam anomalous reflection. Therefore, conversely to the work done in [18], more than one meta-atom is required in the supercell for normal incidence illumination. To address this issue, we propose a fully analytical design methodology for metagratings composed of two meta-atoms and all steps are processed by rigorous theoretical derivations and mathematical calculations. A series of concise formulations are given for the corresponding designs without the need of an optimization procedure. Moreover, auxiliary parameters are introduced to obtain desired power ratio in the diffraction modes. In this way, specular reflection is not only suppressed, but the power ratio can also be freely tailored in beam splitting functionality. Finally, to validate the proposed concept, four types of reflective metagratings are analyzed, designed and numerically demonstrated for an operation under normal incidence: anomalous reflection to 60°, 2) anomalous reflection to 80°, 3) two-way 1:3 beam splitting to +/- 40° 4) three-way 2:1:3 beam splitting to -50°, 0°, + 50°.
2. Theoretical formulation
The metagrating model used for analysis in this work and its coordinate system are shown in Fig. 1. The perfect electric conductor (PEC) ground plane is located at z = 0 on the z-axis, and the periodic array of wires is located at z = -h. The period of the array is A and each period contains two cylindrical wires with radius r, spacing d and load-impedance densities Z1 and Z2, respectively. The medium around the metagrating is air described by permittivity ε and permeability µ. Here, we consider the excitation of the metagrating by a normally incident transverse electric (TE) polarized plane wave along the + z-direction, such that wires of the metagrating carry the polarization currents. The field radiated by these polarization currents and the incident field will be superimposed to form three diffraction beams (-1st, 0th, and +1st), which can be perfectly controlled to achieve various beamforming scenarios.
Fig. 1. Metagrating platform under consideration. A periodic array of polarization line currents (wires) with period A is placed at a distance h over a PEC ground plane. The distance between two line currents in one period is d. The array is excited by a normally incident plane wave with the electric field polarized along the x-direction. (a) 3-D view and (b) top view.
The electric field (considering the incident field plus the reflected field) in air from incident wave shown in Fig. 1 is expressed as
Substituting Eq. (3) into Eq. (2), we can get
In order to simply express Eq. (4) and facilitate subsequent calculations, we define a new function Z (y, z), which is written as
When y ≠ 0 and z ≠ -h, using Poisson summation formula and the Fourier transformation of the Hankel function [32], Eq. (5) can be expressed as
The total electric field of the metagrating includes the electric field (considering the incident field plus the reflected field) in air from incident wave and the radiated field of the line currents array. Thus, combining Eqs. (1), (2) and (5), the total electric field expression of the metagrating in the upper half space (z < 0) can be obtained as
In addition, the diffracted field of the metagrating is the superposition of the radiated field of the line currents array and the reflected field of the incident wave. Therefore, according to Eqs. (2), (6) and the second term of Eq. (1), the diffracted field amplitude of the mth mode of the metagrating can be expressed as
Next, we consider the power ratio issue of the three diffraction modes of the metagrating in this work. If we adopt the conventional method, a set of equations can be obtained by assigning Em according to different m, but at this time, the corresponding polarization currents expression cannot be solved by these equations. So let's ignore the power ratio problem here and consider a simpler single-beam design first, assuming that the +1st diffraction mode is chosen here, then we have
Then we introduce the auxiliary parameters φ1 and φ2 in the form of natural exponents and rewrite the above equations as
It can be found that the right-hand side of the equation system is all 0, which allows us to solve the currents. Here, the parameter φi (i = 1 or 2) is defined as a mathematical level constant that is introduced to enable expanding Eqs. (10) and (11), which can be used to control the power ratio of the three diffraction modes of the metagrating. The case φ1 = 0 indicates that the power of the 0th diffraction mode is zero, and φ2 = ξ-1 or ξ1 indicates that the power of the -1st or 1st diffraction mode is zero. When φ1 and φ2 do not take the three values mentioned above, it means that the power of 0th, -1st and 1st diffraction modes are all not equal to zero. Therefore, we can control the power ratio of the three diffraction modes through the two parameters φ1 and φ2. By solving Eqs. (12) and (13), the expressions of the two line currents per supercell can be derived as
It is important to note that when completing the final metagrating design, the values around the height value h that make the current infinite need to be rounded off, i.e. the value of h that satisfies sin(kh) = 0. Then, according to Ohm’s law given as $E_x^{\textrm{tot}}(y, - h) = {Z_i}{I_i}$ and the total electric field of the metagrating in Eq. (8), we can obtain the corresponding impedance densities of the two meta-atoms per supercell
From Eq. (1), we can write the excitation field at point (0, -h) as
Substituting the Eqs. (14) and (17) into Eqs. (15) and (16). We can get the expanded representations of the two impedance densities
Next, by substituting the impedance density coordinate function defined in Eq. (6) into the above two equations, Eqs. (18) and (19) can be further expanded as
Generally, a very good convergence is obtained with m = 1000 for the calculation of Zi (i = 1 or 2).
For the wavefront manipulation, we need to ensure that the metagrating is passive and lossless so as to target the highest possible efficiency. For this, the impedance density must be purely reactive. Hence, we analyze the mathematical components of the impedance densities Z1 and Z2 to obtain the expressions with specific real part Re{Zi} and imaginary part Im{Zi} (i = 1 or 2) of the impedance densities. The expressions of real and imaginary parts of the impedance densities can be separated as follows
- (1) Real parts:$$\begin{aligned} &{\rm{Re}} [{Z_1}] = \frac{{2\eta }}{A}{\sin ^2}(kh)\sin \left( {\frac{{{\varphi_\textrm{2}}d}}{2}} \right)\left[ {2\sin \left( {\frac{{{\varphi_\textrm{2}}d}}{2} + {\varphi_\textrm{1}}} \right) - \sin \left( {\frac{{{\varphi_\textrm{2}}d}}{2}} \right)} \right]\\ &\quad + \frac{\eta }{{2A}}\sin ({\varphi _\textrm{2}}d)\left[ {\sin (2kh) + 2\sin (2{\beta_1}h)\frac{{\cos ({\xi_1}d)}}{{\cos {\theta_\textrm{1}}}}} \right]\\ &\quad + \frac{{2\eta }}{A}[{\cos ({\varphi_\textrm{2}}d)\cos ({\xi_1}d) - 1} ]\frac{{{{\sin }^2}({\beta _1}h)}}{{\cos {\theta _\textrm{1}}}}\textrm{ + }\sin ({\varphi _\textrm{2}}d)\frac{{k\eta }}{A}\sum\limits_{m = 2}^\infty {\frac{{1 - {e^{ - 2{\alpha _m}h}}}}{{{\alpha _m}}}\cos ({\xi _m}d)} \end{aligned}, $$$$\begin{aligned} &{\rm{Re}} [{Z_\textrm{2}}] = \frac{{2\eta }}{A}{\sin ^2}(kh)\sin \left( {\frac{{\textrm{ - }{\varphi_\textrm{2}}d}}{2}} \right)\left[ {2\sin \left( {\frac{{\textrm{ - }{\varphi_\textrm{2}}d}}{2} + {\varphi_\textrm{1}}} \right) - \sin \left( {\frac{{\textrm{ - }{\varphi_\textrm{2}}d}}{2}} \right)} \right]\\ &\quad + \frac{\eta }{{2A}}\sin (\textrm{ - }{\varphi _\textrm{2}}d)\left[ {\sin (2kh) + 2\sin (2{\beta_1}h)\frac{{\cos ({\xi_1}d)}}{{\cos {\theta_\textrm{1}}}}} \right]\\ &\quad + \frac{{2\eta }}{A}[{\cos (\textrm{ - }{\varphi_\textrm{2}}d)\cos ({\xi_1}d) - 1} ]\frac{{{{\sin }^2}({\beta _1}h)}}{{\cos {\theta _\textrm{1}}}}\textrm{ + }\sin (\textrm{ - }{\varphi _\textrm{2}}d)\frac{{k\eta }}{A}\sum\limits_{m = 2}^\infty {\frac{{1 - {e^{ - 2{\alpha _m}h}}}}{{{\alpha _m}}}\cos ({\xi _m}d)} \end{aligned}, $$where θ1 is the reflection angle of +1st diffraction mode.
- (2) Imaginary parts:$$\begin{aligned} &{\mathop{\rm Im}\nolimits} [{Z_1}] = \frac{{4\eta }}{A}{\sin ^2}(kh)\sin (\frac{{{\varphi _2}d}}{2})\cos (\frac{{{\varphi _2}d}}{2} + {\varphi _1}) + \frac{\eta }{\lambda }\ln (\frac{{2\pi r}}{A})\\ &\quad - \frac{\eta }{A}\sin ({\varphi _2}d)\left[ {{{\sin }^2}(kh) + 2{{\sin }^2}({\beta_1}h)\frac{{\cos ({\xi_1}d)}}{{\cos {\theta_\textrm{1}}}}} \right] + \frac{\eta }{A}G\textrm{(}d,h\textrm{)} \end{aligned}, $$$$\begin{aligned} &{\mathop{\rm Im}\nolimits} [{Z_2}] ={-} \frac{{4\eta }}{A}{\sin ^2}(kh)\sin (\frac{{{\varphi _2}d}}{2})\cos (\frac{{{\varphi _2}d}}{2} - {\varphi _1}) + \frac{\eta }{\lambda }\ln (\frac{{2\pi r}}{A})\\ &\quad + \frac{\eta }{A}\sin ({\varphi _2}d)\left[ {{{\sin }^2}(kh) + 2{{\sin }^2}({\beta_1}h)\frac{{\cos ({\xi_1}d)}}{{\cos {\theta_\textrm{1}}}}} \right] + \frac{\eta }{A}G\textrm{(}d,h\textrm{)} \end{aligned}, $$where$$\begin{aligned} &G\textrm{(}d,h\textrm{)} ={-} \sin (2kh){\sin ^2}(\frac{{{\varphi _2}d}}{2}) + [\cos ({\varphi _2}d)\cos ({\xi _1}d) - 1]\frac{{\sin (2{\beta _1}h)}}{{\cos {\theta _\textrm{1}}}}\\ &\quad + \frac{{kA}}{{2\pi }}\textrm{ + }k\sum\limits_{m = 2}^\infty {\left\{ {\frac{{1 - {e^{ - 2{\alpha_m}h}}}}{{{\alpha_m}}}[{\cos ({\varphi_2}d)\cos ({\xi_m}d) - 1} ]+ \frac{A}{{2\pi m}}} \right\}} \end{aligned}. $$
Here the conditions Re{Z1} = 0 and Re{Z2} = 0 are converted to conditions Re{Z1} +Re{Z2} = 0 and Re{Z1} - Re{Z2} = 0 equivalently such that the conditions ensuring the real part of the impedance densities to be zero can be derived as
$$\frac{{{{\sin }^2}(kh)}}{{{{\sin }^2}(\beta h)}}(2\cos {\varphi _1} - 1) - \frac{{1 - \cos ({\varphi _2}d)\cos ({\xi _1}d)}}{{\cos {\theta _1}{{\sin }^2}({{\varphi_2}d/2} )}} = 0, $$$$\scalebox{0.9}{$\displaystyle\sin ({\varphi _2}d)\left[ {4{{\sin }^2}(kh)\sin {\varphi_1} + 2\sin (2{\beta_1}h)\frac{{\cos ({\xi_1}d)}}{{\cos {\theta_1}}} + \sin (2kh)\textrm{ + 2}k\sum\limits_{m = 2}^\infty {\frac{{1 - {e^{ - 2{\alpha_m}h}}}}{{{\alpha_m}}}\cos ({\xi_m}d)} } \right] = 0$}$$At this point, the analytical expressions of the impedance densities of the metagrating, as well as their passive and lossless conditional equations have been obtained. Next, we will specifically discuss how to control the power distribution of each diffraction mode according to the introduced auxiliary parameters φ1 and φ2. For the power ratio of each diffraction mode in the metagrating, it should satisfy the following conditions [1]
$${P_m} = \frac{{{{|{{E_m}} |}^2}{\beta _m}}}{{{{|{{E_{\textrm{inc}}}} |}^2}{\beta _0}}},\textrm{ }\sum\limits_m {{P_m}} = 1, $$where Pm is the power fraction of the mth diffraction mode. When m = 0, substituting Eq. (9) into Eq. (29) leads to$${P_0} = \frac{{{{|{{E_0}} |}^2}{\beta _0}}}{{{{|{{E_{\textrm{inc}}}} |}^2}{\beta _0}}} = \frac{{{{\left|{ - \frac{{k\eta }}{{2A}}\left( {\frac{{1 - {e^{ - j{\beta_0}2h}}}}{{{\beta_0}}}} \right)({I_1} + {I_2}{e^{j{\xi_0}d}}) - {E_{\textrm{inc}}}{e^{ - jkh}}} \right|}^2}}}{{{{|{{E_{\textrm{inc}}}} |}^2}}}. $$Further substituting Eq. (14) into Eq. (30), and after simplification, we have
When m = -1, substituting Eq. (9) into Eq. (29) in the same way, we obtain
$${P_{ - 1}} = \frac{{{{|{{E_{ - 1}}} |}^2}{\beta _{ - 1}}}}{{{{|{{E_{\textrm{inc}}}} |}^2}{\beta _0}}} = {\left( {\frac{{k\eta }}{{2A{\beta_{ - 1}}}}} \right)^2}\frac{{{{|{({1 - {e^{ - j{\beta_{ - 1}}2h}}} )({I_1} + {I_2}{e^{j{\xi_{ - 1}}d}})} |}^2}{\beta _{ - 1}}}}{{{{|{{E_{\textrm{inc}}}} |}^2}{\beta _0}}}. $$Substituting Eq. (14) into Eq. (32), we can get
$${P_{ - 1}} = \frac{1}{{\cos {\theta _1}}}{\left[ {\frac{{\sin ({\beta_1}h)}}{{\sin (kh)}}} \right]^2}\left[ {\frac{{1 - \cos ({\varphi_2} + {\xi_1})d}}{{1 - \cos ({\varphi_2}d)}}} \right]. $$Further substituting Eq. (27) into Eq. (33) leads to
$${P_{ - 1}} = \left( {\cos {\varphi_1} - \frac{1}{2}} \right)\left[ {\frac{{1 - \cos ({\varphi_2} + {\xi_1})d}}{{1 - \cos ({\varphi_2}d)\cos ({\xi_1}d)}}} \right]. $$According to the symmetry of the -1st diffraction mode and the +1st diffraction mode, the same method can be used to obtain the expression of the power of the +1st diffraction mode as
For the power ratio of -1st, 0th, and +1st diffraction modes, it can be summarized as a: b: 1. The power of the +1st order diffraction mode is taken as the standard and normalization is then performed. Then, according to the expression in Eq. (31), we have
Hence, we can calculate the value of parameter φ1 from Eq. (36). According to Eqs. (34) and (35), we have
Then, we can obtain the values of parameters φ2, d and h through simultaneous Eqs. (27), (28) and (37), (38). Then substituting the calculated parameters φ1, φ2, d and h into Eqs. (20) and (21), we can obtain the required impedance densities Z1 and Z2 of the metagrating.
3. Design of the metagratings and numerical performances
Since the impedance densities calculated for the four scenarios of this study are all negative (as it will be seen in the following results), we use a microstrip capacitor structure and the corresponding relationship between the cylindrical wire and the microstrip capacitor is shown in Fig. 2.
Fig. 2. The equivalent transformation from the cylindrical wire to the strip capacitor. The fixed geometrical parameters are: w = 0.1 mm, g = 0.1 mm, t = 0.018 mm and B = 3 mm. The other parameters (L1, L2, d and A) are variables.
The equivalent relationship between the wire width w of the capacitor and the radius r of the cylindrical wire is w = 4r [18,24], and the length of the strip capacitor L can be determined from [25]
Next, we use the above proposed analytical design methodology to design and simulate metagratings at an operating frequency of 10 GHz using the commercial simulation software HFSS from ANSYS, and copper with conductivity σ = 5.8 × 107 S/m and thickness t = 0.018 mm, is used for the metal strips. We consider designing four metagratings operating under normal incidence, each showing different feature: (i) beam steering with a reflection angle of 60°, (ii) beam steering with a reflection angle of 80°, (iii) two-way beam splitting with reflection angles of -40° and 40° with power ratio of 1:3 and (iv) three-way beam splitting with reflection angles of -50°, 0° and 50° with power ratio of 2:1:3. Their specific calculated structural parameters are given in Table 1 (the parameters A and B of the supercell are calculated as A = λ/sinθ1 (θ1 is the angle of ±1st diffraction mode), and B = λ/10 = 3 mm).
Table 1. Design parameters of the four types metagratings
Simulations are performed on one supercell of the metagrating to verify the power distribution in the different diffraction orders. In this simulation scenario, periodic boundary conditions are applied and a Floquet port is used to illuminate the supercell. Also, in order to observe the far-field radiation pattern, full-wave simulations using a plane wave incidence as illumination have been performed on an array of 20 supercells along the y-axis. The simulation results of the four metagratings are shown in Fig. 3, where the power percentage scattered in a given diffraction mode is marked next to each beam. It should be noted that in the far-field simulation, metal losses are not considered and the corresponding power ratio calculations per diffraction mode is calculated as [25]
Fig. 3. Simulation results of the four types of metagratings where normalized power and real part of electric field are shown. (a) anomalous reflection to 60°, (b) anomalous reflection to 80°, (c) two-way 1:3 beam splitting to +/- 40° (d) three-way 2:1:3 beam splitting to -50°, 0°, + 50°.
Figure 3(a) presents the normalized power of the beam steering at 60°. The efficiency of the +1st diffraction mode is equal to 96%, and the remaining 4% is caused by the specular reflection as well as the absorption due to conductor resistance. In the far-field simulation, the 1.8% of the power loss is due to a small amount of diffraction towards -1st and 0th modes. Figures 3(b), 3(c) and 3(d) show, respectively, the simulation results of the beam steering at 80°, beam splitting at -40°/+40° and beam splitting at -50°/0°/+50°. The losses of these remaining three metagrating supercell lie in the range 1%-2%. The conductor losses vary mainly because of the different heights h above the ground plane that correspond to different current amplitudes, as given in Eq. (14). In general, such conductor losses should not exceed 5%. The simulation results of the four metagrating’s supercells are consistent with the theoretical predictions, indicating that the proposed theoretical design is valid.
4. Conclusion
A fully analytical design methodology for normal-incidence-beamforming metagratings containing low number of meta-atoms has been proposed. A closed form expression of load impedance has been given in formulas (20) and (21), which can be used to directly determine the required design parameters in the design. Meanwhile, the individual parameters are also clear at a glance in the expression for impedance, making it easy to explore its deeper physical meaning subsequently. For designing highly efficient metagrating, Eqs. (27) and (28) can precisely ensure the condition of passive and lossless structures. Finally, the exact Eqs. (36)–(38) for the power ratio of each diffraction mode of the metagrating has also been given, which allows the engineering between the individual diffraction beams of the metagrating to be performed easily. These formulations derived by rigorous calculations provide a solid theoretical tool for future metagratings applications in large-angle beam steering antennas or reflector plates. For a practical implementation of the metagrating structure, a dielectric substrate supporting the meta-atoms above the metal ground plane will be necessary. In such case, the Fresnel reflection coefficient generated by the introduction of the dielectric substrate needs to be taken into account in the expression of the electric field. It should be pointed out that the proposed methodology can only be applied to metagratings whose period contains only a low number of meta-atoms. Indeed, in metagratings with high number of diffraction orders, the calculations become too complex to be done in a full analytical manner and therefore numerical optimization cannot be avoided.
Appendix
When y = 0 and z = -h, Eq. (6) can be written as
In this case, when n = 0, the Hankel function $H_0^{(2)}(k|{nA} |) = 1 + j\infty$, obviously is not convergent. Studies show that $H_0^{(2)}(|x |)$ is convergent when the value of x is small [19,32], for e.g., $H_0^{(2)}({10^{ - 10}}) = 1 + j14.7325$. Thus, we can use the position on the surface of the metal-wire instead of the actual position of (y, z) = (0, -h). Here, we use $H_0^{(2)}(kr)$ to approximate $H_0^{(2)}(k|{nA} |)$ with n = 0, where r is the effective radius of the metal wires. Then Eq. (41) needs to be rewritten as
According to the series expansion of Hankel function using the small argument approximation [19,32], we have
Funding
China Scholarship Council (202206280099).
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 authors upon reasonable request.
References
1. E. G. Loewen and P. Evgeny, Diffraction Gratings and Applications (CRC Press, 2018).
2. G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98(12), 123102 (2005). [CrossRef]
3. P. M. Blanchard, D. J. Fisher, S. C. Woods, and A. H. Greenaway, “Phase-diversity wave-front sensing with a distorted diffraction grating,” Appl. Opt. 39(35), 6649 (2000). [CrossRef]
4. T. Itoh and R. Mittra, “An analytical study of the echelette grating with application to open resonators,” IEEE Trans. Microw. Theory Techn. 17(6), 319–327 (1969). [CrossRef]
5. T. Fujita, H. Nishihara, and J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7(12), 578 (1982). [CrossRef]
6. C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012). [CrossRef]
7. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef]
8. S. Taravati and G. V. Eleftheriades, “Full-duplex reflective beamsteering metasurface featuring magnetless nonreciprocal amplification,” Nat. Commun. 12(1), 4414 (2021). [CrossRef]
9. M. Lin, C. Liu, J. Yi, Z. H. Jiang, X. Chen, H. X. Xu, and S. N. Burokur, “Chirality-intrigged spin-selective metasurface and applications in generating orbital angular momentum,” IEEE Trans. Antenn. Propag. 70(6), 4549–4557 (2022). [CrossRef]
10. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
11. C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110(19), 197401 (2013). [CrossRef]
12. N. Mohammadi Estakhri and A. Alù, “Wave-front transformation with gradient metasurfaces,” Phys. Rev. X 6(4), 041008 (2016). [CrossRef]
13. Y. Ra’di, D. L. Sounas, and A. Alù, “Metagratings: beyond the limits of graded metasurfaces for wave front control,” Phys. Rev. Lett. 119(6), 067404 (2017). [CrossRef]
14. Y. Ra’di and A. Alù, “Metagratings for efficient wavefront manipulation,” IEEE Photonics J. 14(1), 1–13 (2022). [CrossRef]
15. E. Panagiotidis, E. Almpanis, N. Stefanou, and N. Papanikolaou, “Multipolar interactions in Si sphere metagratings,” J. Appl. Phys. 128(9), 093103 (2020). [CrossRef]
16. H. Chalabi, Y. Ra’di, D. L. Sounas, and A. Alù, “Efficient anomalous reflection through near-field interactions in metasurfaces,” Phys. Rev. B 96(7), 075432 (2017). [CrossRef]
17. Y. Ra’di and A. Alù, “Reconfigurable metagratings,” ACS Photonics 5(5), 1779–1785 (2018). [CrossRef]
18. O. Rabinovich and A. Epstein, “Analytical design of printed circuit board (PCB) metagratings for perfect anomalous reflection,” IEEE Trans. Antenn. Propag. 66(8), 4086–4095 (2018). [CrossRef]
19. A. Epstein and O. Rabinovich, “Unveiling the properties of metagratings via a detailed analytical model for synthesis and analysis,” Phys. Rev. Appl. 8(5), 054037 (2017). [CrossRef]
20. F. Boust, T. Lepetit, and S. N. Burokur, “Metagrating absorber: design and implementation,” Opt. Lett. 47(20), 5305 (2022). [CrossRef]
21. Z. Tan, J. Yi, Q. Cheng, and S. N. Burokur, “Design of perfect absorber based on metagrating: theory and experiment,” IEEE Trans. Antenn. Propag. 71(2), 1832-1842 (2023), [CrossRef] .
22. H. Rajabalipanah, A. Momeni, M. Rahmanzadeh, A. Abdolali, and R. Fleury, “Parallel wave-based analog computing using metagratings,” Nanophotonics 11(8), 1561–1571 (2022). [CrossRef]
23. O. Rabinovich and A. Epstein, “Nonradiative subdiffraction near-field patterns using metagratings,” Appl. Phys. Lett. 118(13), 131105 (2021). [CrossRef]
24. V. Popov, F. Boust, and S. N. Burokur, “Controlling diffraction patterns with metagratings,” Phys. Rev. Appl. 10(1), 011002 (2018). [CrossRef]
25. V. Popov, F. Boust, and S. N. Burokur, “Constructing the near field and far field with reactive metagratings: study on the degrees of freedom,” Phys. Rev. Appl. 11(2), 024074 (2019). [CrossRef]
26. V. Popov, M. Yakovleva, F. Boust, J.-L. Pelouard, F. Pardo, and S. N. Burokur, “Designing metagratings via local periodic approximation: from microwaves to infrared,” Phys. Rev. Appl. 11(4), 044054 (2019). [CrossRef]
27. V. Popov, F. Boust, and S. N. Burokur, “Beamforming with metagratings at microwave frequencies: design procedure and experimental demonstration,” IEEE Trans. Antenn. Propag. 68(3), 1533–1541 (2020). [CrossRef]
28. G. Xu, S. V. Hum, and G. V. Eleftheriades, “Dual-band reflective metagratings with interleaved meta-wires,” IEEE Trans. Antennas Propag. 69(4), 2181–2193 (2021). [CrossRef]
29. A. Casolaro, A. Toscano, A. Alù, and F. Bilotti, “Dynamic beam steering with reconfigurable metagratings,” IEEE Trans. Antenn. Propag. 68(3), 1542–1552 (2020). [CrossRef]
30. O. Rabinovich and A. Epstein, “Arbitrary diffraction engineering with multilayered multielement metagratings,” IEEE Trans. Antenn. Propag. 68(3), 1553–1568 (2020). [CrossRef]
31. G. Xu, G. V. Eleftheriades, and S. V. Hum, “Analysis and design of general printed circuit board metagratings with an equivalent circuit model approach,” IEEE Trans. Antenn. Propag. 69(8), 4657–4669 (2021). [CrossRef]
32. S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).
