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Abstract
Metagrating is a new concept for wavefront manipulation that, unlike phase gradient metasurfaces, does not suffer from low efficiency and also has a less complicated fabrication process. In this paper, a compound metallic grating (a periodic metallic structure with more than one slit in each period) is proposed for anomalous reflection. We propose an analytical method for analyzing the electromagnetic response of this grating. Closed-form and analytical expressions are presented for the reflection coefficients of zeroth diffracted order and also higher diffracted orders. The proposed method is verified against full-wave simulations and the results are in excellent agreement. Thanks to the geometrical asymmetry of compound metallic grating, it can be used for designing anomalous reflection at the normal incidence. Given analytical expressions for reflection coefficients, we design a perfect anomalous reflector for a TM polarized plane wave via transferring all the incident power to ( − 1) diffraction order . The structure designed in this study has an unprecedented near-to-unitary efficiency of 99.9%. Finally, a multi-element compound metallic grating is proposed for reflecting the normal incidence to angles of below 30°, which is a challenging accomplishment. This excellent performance of compound metallic grating shows its high potential for microwave and terahertz wavefront manipulation applications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The anomalous reflection/refraction is an interesting subset of wave manipulations owing to its important role in laser, optics, and so on [1–3]. For a long time, this attractive phenomenon has been realized by means of reflect arrays [4]. Nowadays, due to the development of nanotechnology, metasurfaces are common elements for achieving anomalous reflection [5–7].
Metasurfaces, two-dimensional artificial structures used for manipulating the propagation of electromagnetic (EM) waves, have attracted great attention because of their numerous applications [8–18]. More specifically, phase gradient metasurface (PGM) can provide exceptional phenomena such as surface wave conversion [19], vortex beam generation [20], and beam focusing [21]. Also, PGM can realize the anomalous reflection based on generalized Snell’s law, which is achieved by the inhomogeneous phase profile of metasurfaces [22]. Highly efficient anomalous reflection is a strict requirement for the above-mentioned applications. However, metasurfaces have limited efficiencies unless they are active and extremely local [23–25]. Moreover, fabrication of metasurfaces has significant difficulties.
Recently Ra’di et al. have introduced the concept of metagratings, which enables manipulations of EM waves with unitary efficiency [26]. In contrast to metasurfaces, metagratings do not need deeply subwavelength elements, making their fabrication process less complicated. The concept of metagrating is similar to the diffraction grating which has been one of the attractive research topics in the past decades [27–30]. The design principles of metagratings are based on Floquet-Bloch (FB) theory. Based on FB theory, when a plane wave impinges on a periodic structure, a discrete set of diffracted waves can be generated that some of them are propagating and others are evanescent. The number of propagating and evanescent waves is determined by the period of the structure and the angle of incidence. To achieve anomalous reflection, all the propagating modes should be vanished (including the specular mode) except one of them and thus all the power is transferred to that propagating non-specular FB mode.
Inspired by this idea, several anomalous reflectors have been proposed but most of them are designed through numerical simulations [31–35]. Although some analytical methods are proposed, all of them have some problems that restrict their performance. For example, [36] considered a metagrating formed by pairs of isotropic dielectric rods suspended in the free space above a perfect electric conductor (PEC) ground, which is not practical for implementation. Furthermore, the analytical method presented in [37–39], due to its geometric symmetry, cannot be used in the case of normal incidence, which is the most important case in the anomalous reflectors. After all, the power efficiency of the previously proposed metagrating-based anomalous reflectors is limited to $94\%$ [31,37,40]. Moreover, none of the proposed structures provide anomalous reflection below $30^\circ$. This case is more challenging, since $(\pm 2)$ FB modes become propagating. Although, a method of design is proposed based on multiple electric line currents for this case [41], however a practical structure with unitary efficiency has not been presented to the best of the author’s knowledge.
In this paper, we show that achieving unitary efficiency for anomalous reflection is possible using compound metallic gratings (CMGs). A CMG includes finite numbers of slits in each period, which can be used as a reflection grating (consists of slits carved out of a thick metal slab) or transmission grating (connects two separate open regions through groups of slits in a metal slab). In [42], an analytical equivalent-circuit model has been proposed for compound metallic gratings with an arbitrary number of slits per unit. However, higher diffracted orders cannot be calculated in this method. We extend this equivalent-circuit to include higher order diffracted modes in our method. Thanks to this analytical method, we design a metagrating with near-to-unitary efficiency ($99.9\%$) at normal incidence using a CMG. We also demonstrate anomalous reflection below $30^\circ$ with an impressive efficiency of $98.5\%$.
2. Analytical method for analysis of CMGs
2.1 Scattering from compound metallic grating
The schematic of CMG, which includes two slits in a period, is illustrated in Fig. 1. Each unit cell consists of two slit regions with widths of $w_2$ and $w_3$ filled with a dielectric medium, the refractive indices of these regions are $n_2$ (region II) and $n_3$ (region III), respectively. The distance between the center of slits is $d$ and the height of the first and second slit is $h_2$ and $h_3$, respectively (Fig. 1). The period of the array is $P$ in the x-direction. Also, the structure is assumed to be infinite along the y-direction ($\partial /\partial y\sim 0$). It should also be noted that the time-harmonic of the form $\exp (j{\omega }t)$ is assumed throughout this paper.
Fig. 1. The structure of CMG with two slits per period. (a)3D view (b)x-z plane. CMG is illuminated by an oblique TM polarized wave with angle $\theta _i$
Consider that the structure is illuminated by an oblique transverse-magnetic (TM) polarized plane wave (the magnetic field is along the $y$ direction) propagating along the z-direction with incidence angle of $\theta _i$. Based on FB theorem, the total field in the region $z>0$ is expressed by the Rayleigh expansion [43]
In regions II and III, we assume slits are in single-mode regime and thus only transverse electromagnetic (TEM) mode is propagation in these regions. Hence, the magnetic and electric fields in regions I and II can be written as
Since we assume that inside the slits only the fundamental mode is dominant and higher-order modes are in cut-off, the validity of this method is limited to the frequency range $\lambda >\max [2w_i n_i] ~~(i=2,3)$ [43].
Now, by applying the appropriate boundary conditions at $z=0$ for the electric fields (the continuity of the tangential electric field at every point of the unit cell) using 1b,5b and 6b we have
These calculations can be readily generalized to the case of multiple CMGs (a CMG that includes multi-slit in each unit cell) following a similar approach.
For TE polarization, like the TM polarization of incident wave, first, we define the electric and magnetic fields in each region, except that in regions II and III, we only take into account the first TE mode of the parallel plate waveguide. Then, by applying boundary conditions, reflection coefficients can be derived [42]. For the sake of brevity, these calculations are not included here.
In all the above equations, we used the PEC approximation for CMG analysis. Therefore, at higher frequencies close to optical regime, our model is not valid. For adding the effects of ohmic losses of the metallic surfaces to our model, we apply the approach used in [44] and we assume good conductor approximation and strong skin effect conditions where each slit is considered a lossy parallel-plate waveguide with a complex wavenumber given by [45]:
where $\delta _s=\sqrt {2/\omega \mu _0 \sigma }$ is the skin penetration depth into the metallic surfaces and $\sigma$ is the is the conductivity of metal. These calculations can be readily generalized to the cases of lossy metals by substituting 16 in 7.2.2 Numerical results
In this section, we validate and verify the accuracy of the proposed circuit model through some numerical examples. We demonstrate that our proposed model produce very accurate results as long as $\lambda >\max [2w_in_i]$.
Consider a CMG in accordance with Fig. 1. Here, the normalized frequency is defined as $\omega _n=P/\lambda$ and the CMG parameters are set to $w_2=0.2P, w_3=0.15P$, $h_2=1.25P$, $h_3=1.625P$, $d=0.25P$, $n_1=n_2=1$ and $n_3=1.5$. The diffraction efficiency of zeroth (specular) diffracted order obtained by the analytical method for different incident angles is plotted in Fig. 2(a). To validate the proposed analytical method in the previous subsections, a full-wave simulation is performed using the finite integration technique (FIT) in CST Microwave Studio 2019. Also, the first and the second diffraction efficiencies of the structure are plotted in Fig. 2(b) for the normal incidence. Excellent agreement is observed between the FIT results and our analytical method, demonstrating the accuracy of the proposed model. The difference between $DE_1$ and $DE_{-1}$ can be observed from Fig. 2(b). This difference stems from the geometrical asymmetry of the structure along the $x$-direction, which can be highly useful in the design of metagratings and anomalous reflectors [46].
Fig. 2. Comparing the results of proposed method with full wave simulations: (a)The diffraction efficiency of zeroth-order mode for different incident angles: $\theta _i=0^\circ , 30^\circ$ and $~60^\circ$. (b)The diffraction efficiency of higher orders at normal incidence. The CMG parameters are assumed as $w_2=0.2P, w_3=0.15P, h_2=1.25P, h_3=1.625P, d=0.25P, n_1=n_2=1$ and $n_3=1.5$.
3. Perfect anomalous reflector: design and simulation
3.1 Design of perfect anomalous reflectors
In this section, we present an analytical method for the design of metagratings with anomalous reflection achieving unity efficiency. Based on the FB theorem, a discrete set of modes can be diffracted when a plane wave impinges to a periodic structure. As shown in Fig. 1, except the specular mode (zero-order), all the FB modes diffract to various angles that are not equal with the angle of the incident wave. If all the power of incident wave transfer to a propagating FB mode, the perfect anomalous reflection can be achieved. When the angle of the incident wave is normal, for a perfect anomalous reflection, the structure must have geometrical asymmetry in $x$-direction; otherwise, in the best situation, the efficiency of anomalous reflection is $50\%$ [26].
The proposed CMG structure has geometrical asymmetry in $x$-direction and thus can be used in metagrating applications. For the simplification of design process, we make two assumptions: 1- normal incidence, 2- $0$ (specular) and ($\pm 1$) orders are the only propagating FB modes and higher-order modes are evanescent (at normal incidence this condition can be satisfied with $P<2\lambda$). Our goal is to vanish (0) and (1) modes and transfer all the power to $(-1)$ mode. First, we specify the period of the CMG based on the desired angle of anomalously reflected wave:
obtained from 2a. For more simplicity in fabrication process, we assume that both slits are filled with air ($n_1=n_2=n_3=1$). As the main purpose, the (0) and (1) FB modes of the proposed structure should be zero. According to passivity condition and since the higher order modes(i.e., $m\geqslant 2$) are evanescent, by eliminating the $DE_0$ and $DE_{1}$, we achieve unitary efficiency for the $DE_{-1}$. Based on the theoretical formulation presented in the previous section, the genetic algorithm is utilized to minimize the $DE_0$ and $DE_{1}$ of the anomalous reflector. Hence, we define the cost function as $DE^2_{0}+DE^2_{1}$ in the desired frequency. The height and width of slits and also the distance between the centers of two slits are the variables in the optimization process. It should be noted that due to the periodic nature of 11 and 14, the height of slits always can be chosen from the interval $[0,\lambda /(2n_i)]$.Using the above-mentioned design process we designed an anomalous reflector with near-to-unitary efficiency that reflects the incident wave to a direction with a $-50^\circ$ reflection angle. The parameters of the designed CMG are $w_2=0.21P, w_3=0.08P, h_2=0.21P, h_3=0.14P$ and $d=0.36P$. The diffraction efficiency of the designed CMG is shown in Fig. 3(a). It can be found from Fig. 3, that almost all the power ($99.9\%$) couples to the (−1) FB mode in the desired frequency $\omega _n=1.3$. This efficiency is a remarkable achievement in comparison with the previously published studies. The power efficiencies of anomalous reflection in [36], [37], [31], [32], and [40] are $94\%, 94.1\% ,94\%$, $82\%$ and $74.8\%$, respectively. The distribution of the reflected electric fields at the design frequency ($\omega _n=1.3$) is plotted in Fig. 3(b) which clearly shows anomalous reflection.
Fig. 3. (a) Diffraction efficiency of the designed anomalous reflector. (b)The electrical field distribution for $\omega _n=1.3$. The CMG parameters are designed as $w_2=0.21P, w_3=0.08P, h_2=0.21P, h_3=0.14P$ and $d=0.36P$ for transferring the incident power to the angle $-50^\circ$. E-field distribution is obtained by CST Microwave Studio.
The CMG can be used for reflecting the normal incidence to a direction with different angles. We extract the parameters of CMG similar to the previously designed anomalous reflector for several $\theta _{-1}$ values in the range $-40^\circ$ to $-80^\circ$. Also, CMG can anomalously reflect oblique incidence electromagnetic waves. As a prototype, we design a metagrating that can reflect an incident wave from $\theta _i=50^\circ$ to $\theta _{-1}=-22.5^\circ$. The optimized structure parameters and the power efficiency of designed metagrating are listed in Table 1. It can be observed from Table 1 that in all of the designed anomalous reflectors, we can achieve near to unitary efficiency. It should be noted that according to 2a, the anomalous reflection occurs in the normalized frequency $\omega _n=P/\lambda =1/|(sin\theta _i - sin\theta _{-1}|)$ for both of the oblique and the normal incidence cases.
Table 1. Optimum parameters for the anomalous reflection using CMG. For simplifying the fabrication process, the slits are filled with air.
For reflecting the normal incidence wave to the angle of below $30^\circ$, based on 17, it is required that $\lambda >2P$. According to 2a and this condition, $\pm 2$ orders of FB modes become propagating. Therefore, the design of this case is more challenging and thus there are little works on this case. In [41], it has been shown that a metagrating having $N$ polarization line currents per unit cell can eliminate the $N-1$ FB modes. However, this work did not propose a practical structure for the realization of anomalous reflection with unitary efficiency at angles below $30^\circ$.
For the design of an anomalous reflector at $\theta _{-1}=-25^\circ$ for the normal incidence waves, based on 17, periodicity must be chosen as $2.36\lambda$. It is noteworthy that similar to previous cases, our goal is to vanish all FB modes except $m=-1$. The $(0), (\pm 1$) and ($\pm 2)$ FB modes are propagating for a structure with a period of $2.36\lambda$ . In this case, according to [41], the proposed structure of Fig. 1, which has two elements per period, cannot be used. Hence, we propose a multi-element CMG as depicted in Fig. 4. The proposed structure has four slits in a period. Again for easier fabrication, we assume the slits are filled with air.
Fig. 4. The structure of multiple CMG with four slits per period. (a)3D view (b)x-z plane. CMG is illuminated by the normal incident
The method of design is similar to the previously designed metagrating. By eliminating $(0), (1),$ and $(\pm 2)$ orders of the FB mode, the parameters of structure in Fig. 4 are designed as $w_2=0.12P$, $w_3=0.06P$, $w_4=0.11P$, $w_5=0.07P$, $h_2=0.11P$, $h_3=0.16P$, $h_4=0.08P$, $h_5=0.06P$, $d_1=0.13P$, $d_2=0.16P$ and $d_3=0.25P$ for reflecting incident wave into $-25^\circ$. The diffraction efficiencies of the propagating FB modes are plotted in Fig. 5(a). As can be observed from Fig. 5(a), the incident power goes to (−1) mode with excellent power efficiency ($98.5\%$). The electrical field distribution plotted in the design frequency demonstrates that the incident power is reflected at the angle of $-25^\circ$ (Fig. 5(b)). E-field distribution is plotted using CST Microwave Studio 2019.
Fig. 5. (a) $DE_0,DE_{\pm 1}$ and $DE_{\pm 2}$ of the designed anomalous reflector for reflected angle of $-25^\circ$, (b)The electrical field distribution at ($\omega _n=2.36$). The multiple CMG parameters are designed as $w_2=0.12P$, $w_3=0.06P$, $w_4=0.11P$, $w_5=0.07P$, $h_2=0.11P$, $h_3=0.16P$, $h_4=0.08P$, $h_5=0.06P$, $d_1=0.13P$, $d_2=0.16P$ and $d_3=0.25P$.
3.2 Proposal for experimental realization
In this subsection, we describe how our anomalous reflector can be realized in the microwave regime. As a proof of concept, we demonstrate the design of a CMG operating at 10 GHz that anomalously reflects the normal incidence wave to the desired direction of $\theta _{-1}=-60^\circ$.
Unlike the previous subsection, we use lossy metal for designing metagrating (copper with conductivity of $\sigma =5.96\times 10^7$ is considered in this design). Based on 17, P must be chosen as 34.64 mm. An optimization has been adopted to eliminate 0 and 1 FB order and the designed structural parameters are $w_2=8.25 mm$, $w_3=4.8 mm$, $h_2=9 mm$, $h_3=4.8 mm$, $d=13 mm$, and $n_1=n_2=n_3=1$. The diffraction efficiency of the optimized structure is plotted in Fig. 6(a). Perfect anomalous reflection is observed at 10 GHz (99.9 % power efficiency).
Fig. 6. Proposed metagrating that anomalously reflects the normal incidence wave to the angle $-60^\circ$. The designed CMG parameters are $P=34.64 mm$, $w_2=8.25 mm$, $w_3=4.8 mm$, $h_2=9 mm$, $h_3=4.8 mm$, $d=13 mm$, $n_1=n_2=n_3=1$ and $\sigma =5.96\times 10^7$. a)The diffraction efficiency of the designed anomalous reflector. (b) Schematic of RCS simulation setup. (c) Bistatic RCS for the CMG (blue curve) compared against that for a PEC sheet of the same size (red curve) at the desired frequency (10 GHz). (d) The far-field scattering patterns of CMG at 10 GHz.
We proceed to simulate the far-field properties of the CMG when it is truncated to a finite size. The physical size of CMG is approximately 300 mm in the y-direction and 415.7 mm in the x-direction (12 unit cell). Figure 6(b) represent a schematic setup for bistatic radar cross section (RCS) calculation. The truncated CMG is subject to a TM-plane electromagnetic wave that is normally incident on the grating plane ($z=0$ plane). The receiver antenna is far enough from metagrating and the RCS calculations are performed under the far-field condition.
Figure 6(c) shows a comparison between RCS of the designed metagrating and that of a metallic plate of the same size as a reference. The bistatic RCS is plotted versus the elevation angle ($\theta$) at 10GHz. The main peak of reflection toward the receiving antenna occurs when the elevation angle $(\theta )$ be in $-60^\circ$ that we expected these results from our analytical method. Moreover, we find that highly-effective suppression of specular reflection can be obtained using the proposed structure. The RCS of the CMG at $\theta =0$ (specular angle) is 23.4 dB less than that of PEC sheet and, on the other hand, at $\theta =-60$ an excess RCS of about 55 dB for the designed CMG is achieved. Hence most of the power is anomalously reflected to the desired direction of $\theta =-60$. Furthermore, we plotted the 3D scattering patterns of the designed metagrating in Fig. 6(d). Anomalous reflection obviously can be observed in Fig. 6(d). In all the above simulations, CST Microwave Studio 2019 was utilized to extract the far-field performance of the CMG.
4. Conclusion
In this paper, CMG was proposed for achieving perfect anomalous reflection as one of the most popular phenomena in the microwave, THz, and optical regime. An analytical method was proposed for analyzing CMG structures where the diffraction efficiency of higher diffracted orders can be calculated. The results of the proposed method were validated against full wave simulations, showing excellent agreement between the two solutions. Then, thanks to the proposed analytical method, perfect anomalous reflectors with virtually unitary power efficiency were presented to reflect the normal incidence wave into angles above $-30^\circ$. Moreover, multi-element CMG was proposed to reflect the normal incidence wave to angles below $-30^\circ$. Our analytical results demonstrated that perfect anomalous reflection could be realized using CMGs, which can be very useful in the context of metagratings and wave manipulation.
Funding
Research Office of Sharif University of Technology.
Disclosures
The authors declare no conflicts of interest.
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