Deterministic Approach to Achieve Broadband Polarization-Independent Diffusive Scatterings Based on Metasurfaces

Transcription

1 Supporting Information for Deterministic Approach to Achieve Broadband Polarization-Independent Diffusive Scatterings Based on Metasurfaces He-Xiu Xu 1,,6 *, Shaojie Ma 1, Xiaohui Ling 1,6, Xiao-Kuan Zhang, Shiwei Tang 5, Tong Cai 1,, Shulin Sun 3, Qiong He 1,4, Lei Zhou 1,4 * 1 State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education) and Physics Department, Fudan University, Shanghai 433, China Air and Missile Defense College, Air force Engineering University, Xi'an, 7151, China 3 Shanghai Engineering Research Center of Ultra-Precision Optical Manufacturing, Green Photonics and Department of Optical Science and Engineering, Fudan University, Shanghai 433, China 4 Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 433, China 5 Department of Physics, Faculty of Science, Ningbo University, Ningbo 31511, China 6 College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 41, China * Corresponding authors: Lei Zhou, phzhou@fudan.edu.cn; He-Xiu Xu, hexiuxu@fudan.edu.cn. This PDF file includes: Page S1-S17, Figure S1-S16 The contents are listed as 1. Derivations of the array-theory formulas Eqs. (3-4)... S. Scattering properties of linear-phase metasurfaces... S3 3. Scattering properties of the focusing-phase subarrays and the criterion to determine the optimal size of a subarray... S4 4. Optimization of the focusing-phase metasurface... S7 5. Scattering properties of a quasi-periodic focusing-phase metasurface... S8 6. Additional FDTD results for focusing-phase metasurfaces... S9 7. FDTD results for a random-phase metasurface... S14 8. Additional experimental results for the focusing-phase metasurfaces... S15 9. Pictures of the experimental setup... S17 S1

2 1. Derivations of the array-theory formulas Eqs. (3-4) In this section, we derive the array-theory formulas for analytically studying the wave-scattering properties of our metasurfaces. Using such a scheme, we can grasp the essential scattering properties of a metasurface without necessarily performing time-consuming full-wave simulations, which can help us retrieve the required phase profile to achieve the best wave-diffusion scatterings. The array theory is suitable to study the case that all antennas (or subarrays in our case) exhibit the same scattering patterns, but are located at different local positions and possess different initial phases. Since our metasurface is composed by a collection of subarrays arranged in some particular coding sequence, we consider the radiation pattern of a typical subarray first. As shown in Eq. () in the main text, a subarray consists of P P antennas arranged in a square lattice, with the antenna located at the (local) position (u,v) possessing an initial phase given by ϕ ( uv foc, ) + ϕ ( ξ ) under LCP excitation. We also note that different antenna inside the subarray exhibits identical scattering field distribution, with the only difference being that their local positions (u,v) and initial phases ϕ foc ( uv, ) + ϕ ( ξ ) are different. Therefore, seen at a far-field point M, the total electric field can be considered as the summation of that radiated from each antenna source. Assuming that the radiation filed from a single antenna is given by E (which depends on the structure of the basic PB meta-atom), we find that the radiation field of a subarray can be approximately written as E = E sub inside subarray antennas = ikma sinθcosφ ikna sinθsin φ i ϕ foc ( ma, na) ik [ rξ + ϕ ( ξ)] E e e e e m n ik [ rξ + ϕ ( ξ)] sub = E F e where ϕ ( ξ ) is a constant value within the ξ -th subarray and r ξ subarray in the global coordinate system. (S1) is the center position of this Equation (S1) clearly tells us that all subarrays exhibit the same radiation patterns, with the only difference being that their local positions (denoted by r ξ ) and their initial phases (denoted by ϕ ( ξ )) are different. To obtain the radiation field of the entire metasurface, we can repeat the same procedure to sum up the contributions from different subarrays, yielding S

3 E total = ik [ rξ + ϕ ( ξ)] ikmpa sinθcosφ iknpa sinθsinφ i ϕ ( M, N) sub M N ξ E sub ( ξ) = E F e sub = E F e e e = E F F sub ξ AF (S) Under the lowest-order approximation, a PB meta-atom can be viewed as two in-plane dipoles orientated along two orthogonal axes exhibiting a π initial-phase difference. Therefore, under the CP-light excitation, we find that the radiation of a single PB meta-atom must exhibit the following angle dependence: E cos θ. Therefore, the scattering pattern of the entire metasurface is determined by S ( θ, φ) Etotal θ, φ cos θ F θ, φ F θ, φ ( ) ( ) ( ) total Sub AF (S3) which is Eq. (3) in the main text.. Scattering properties of linear-phase metasurfaces We employ the array theory developed in last section to study scattering properties of the 1D linear-phase metasurfaces. Suppose that our metasurface only contains one subarray exhibiting the following reflection phase distribution ϕ( x) = ξx within x [ L, L ], we find that 1 L cos 1 L iξ x ik x θ FSub( θφ, ) = dxe e dxe L = L L L sin cos L = = sin c ( ξ k cosθ) L cos ( ξ k θ) ( ξ k θ) L ( ξ cosθ) i k x (S4) and that the scattering pattern is (using Eq. (S3)) ( θφ) = θ ( ξ θ) S, cos sin c k cos L total (S5). For a metasurface with sufficiently large size ( kl 1), the sinc function turns to be a delta function with non-zero value only occurring at the angle determined by cosθ c = ξ k (see radiations patterns shown in Figure S1a). However, for a metasurface with a small size, the term cos θ of a single element dominates the pattern, so that the directivity and the peak intensity of the radiation pattern decay as L decreases (see Figure S1b). S3

4 Figure S1 Theoretically-calculated scattering pattern of a linear-phase subarray at 1 GHz (wavelength of λ=3 mm) with (a) various ξ and (b) various size L. In the former case L=3 mm whereas in the latter case ξ = k. 3. Scattering properties of the focusing-phase subarrays and the criterion to determine the optimal size of a subarray We now employ the array theory to examine the scattering properties of a single focusing-phase subarray with length L. Consider first a subarray exhibiting the following 1D phase distribution with f being the focal length and x [ L, L ] evaluated as ( θ) ( x) k ( ) f x f ϕ = + (S6) S4, we find that its scattering pattern can be ( + ) 1 L ik f x f ikxcosθ = cos θ. (S7) L Stotal dxe e L We numerically evaluated Eq. (S7) for systems with different values of subarray size L, with working wavelength set as λ=3 mm and focal length f = 5 mm. Figure Sa shows that the scattering property of a subarray indeed strongly depends on its size: Whereas only one specular-reflection peak exists in the case of L< L = 7 mm, more and more peaks appear and the scattering to different angle becomes more and more uniform when L> L and L becomes even larger. To appreciate the physical meaning of the critical length L, we depict in Fig. Sb the phase distribution for the case of L= L. Obviously, we find that L= L is the criterion to enable the

5 subarray to exhibit full 36 o phase coverage. Figure S (a) Theoretically-calculated scattering pattern of a 1D focusing-phase metasurface at 1 GHz (wavelength of λ=3 mm) with various L. (b) Phase distribution for the case of L= L. Here, L=7 mm and f=5 mm are fixed. However, the 1D case can only give us the information on θ -dependence of the scattering properties. We now turn to study a L L sized square-shape subarray with a D focusing phase distribution ( x, y) k ( f x y f ) ϕ = + +, in order to fully understand its wave-diffusion characteristics in terms of both θ and φ dependences. Figure S3 depicts the numerically calculated far field scattering pattern for a subarray with L=3 mm, again setting the working wavelength λ=3 mm and focal length f = 5 mm. We find that the scattering pattern of such a subarray exhibits 4 sharp peaks at four φ =45 o, 135 o, 5 o and 315 o, which must be contributed by the corner scatterings of the square-shape subarray and is highly unfavorable for wave-diffusion applications. Such unfavorable directive emissions on the φ -plane can be significantly suppressed if we decrease L, which is quite physical because corner scatterings are diminished when L decreases and completely disappear when L approaches zero (a point), see Figure S4a. As can be seen from Figure S4a, more beams appear on the φ -plane and the distribution becomes more uniform as L decreases. On the other hand, when L is too small (smaller than Lo=7 mm), single beam is observed at θ = o (see Fig. S4b, consistent with Fig. Sa), which is again unfavorable for diffusive scatterings. Compromising these competing effects, we derive the following empirical criterion to determine the optimized length of a subarray, that is, it is such a value to enable the phase variation over the whole subarray to just cover the full 36 o range. S5

6 Figure S3 Theoretically calculated scattering pattern of a D focusing-phase metasurface as a function of azimuth angle φ and elevation angle θ at 1 GHz with focal length of f=5 mm and L=3 mm. Figure S4 Theoretically calculated scattering behavior of a D focusing-phase metasurface at 1 GHz (wavelength of λ=3 mm) with various L at f=5 mm. (a) The totally scattered power (a) 9 S( θ, φ ) dθ and (b) 36 S( θ, φ ) dφ as a function of φ and θ, respectively. To understand the origin of the directive emission peaks on the φ -plane for the square-shape subarray, we construct subarrays with near-circular shapes, and then numerically studied their scattering properties. As shown in Figure S5, compared to their square-shape counterparts of the same sizes, the non-uniform variations of the scattering distributions on the φ -plane are indeed deeply suppressed for a circular-shape subarray. S6

7 Figure S5 Theoretically calculated 3D scattering patterns of different subarrays with nearly circular shapes with diameters of (a) 65.7 mm, (b) 98.5 mm, (c) mm, and (d) mm. Inside these subarrays, the meta-atoms are arranged in a square lattice with lattice constant 1 mm. 4. Optimization of the focusing-phase metasurface The effectiveness and robustness of our approach using focusing-phase metasurface can be further validated by comparisons of phase profile with and without optimization. For simplicity, here we only consider the 1D metasurface of a fixed length. We employ numerical methods to search for the optimized phase distribution on such a metasurface that can achieve the best wave-diffusion performance. Specifically, to describe an arbitrary phase distribution, we assume that the phase distribution is given by a series of Taylor expansions with initial expansion coefficients an chosen to yield a standard focusing phase profile, and then we optimized those coefficients an, based on the criterion that the fluctuation of the angular-dependence of the radiation reaches a minimum. The main optimization process includes the following steps: 1. Define an initial phase distribution which satisfies ( ) ( ) int x k f x f. Fit the focus phase distribution ( x) satisfy ϕ ( ) int Nmax n n,int. n= x = a x 3. Obtain the radiation pattern of a given phase distribution. 4. Define an error function F std ( S) err int ϕ = +. ϕ to obtain the initial Taylor expansion coefficients to =. 5. Randomly vary the strength of an to decrease the error function. 6. Repeat the random variation process with enough time and obtain the optimized S(θ) with desired phase distribution. S7

8 7. Fit the optimized phase distribution using focusing phase of certain focal length fopt. Without losing generality, we choose a metasurface with a size of L= mm, set the working wavelength as λ= mm and the initial focal length as f=5mm, and adopt the maximum Taylor expansion order Nmax=. After 11 times searching iterations, we obtain the final optimized phase distributions and the corresponding radiation patterns, and depict the results in Figure S6. Figure S6 shows that the finally optimized phase distribution looks very much like a focusing-phase profile with fopt=8.7 mm. Even for the initial focusing-phase metasurface with f=5 mm, it also exhibits a good diffusion effect. Figure S6 Optimized results based on Taylor expansion and array theory. (a) Phase distributions and (b) theoretically-calculated scattering patterns. 5. Scattering properties of a quasi-periodic focusing-phase metasurface To examine the intrinsic capability of our strategy in re-distributing the incident energy into various directions, we first calculated the scattered behavior of a quasi-periodic focusing-phase metasurface consisted by periodically repeated 5 5 divergent subarrays under normal incidence of x/y-polarized LP waves and LCP/RCP waves, see Figure S7a. A total of 5 5 basic meta-atoms accommodate an overall square area of 3 3 mm. As shown in Figure S7b, obvious monostatic RCS reduction is inspected from 7 to 18 GHz which coincides well with the bandwidth of out-of-phase orthogonal reflections of PB meta-atoms. The -6 db RCS reduction is immune to polarizations and is observed within 9.5~17.6 GHz, corresponding to a fractional bandwidth of 6%. The three-dimensional (3D) scattering pattern at 13. GHz shown in the inset illustrates that the re-distribution of the incident energy into various directions gives rise to the fairly good backscatter reduction. For further demonstration, we plot in Figures S7c and S7d the D scattering patterns at S8

9 several representative frequencies. As discussed previously, the scattering energy from the metasurface is dispersed toward φ = o, 45 o, 9 o, 135 o, 18 o, 5 o, 7 o, and 315 o, which is attributable to the square shape of the focusing subarrays as discussed in the main text. To sum up, the focusing metasuface is capable of intrinsically diffusing the reflection waves toward numerous directions even without coding manner. Most importantly, such diffusion is independent on wave polarizations and is also preserved for focusing metasurfaces with convergent subarrays. Figure S7 FDTD calculated far-field scattering performance of the quasi-periodic metasurface consisted by 5 5 divergent focusing-phase subarrays. (a) Metasurface layout with focusing-phase subarray shown in the inset. (b) Monostatic RCS reduction under normal incidence of different polarizations, the inset depicts the 3D scattering pattern at 13. GHz. (c) D scattering intensity across φ and θ angle at 13. GHz. (d) 3D scattering patterns at several representative frequencies. 6. Additional FDTD results for focusing-phase metasurfaces Figure S8 shows the phase distributions of the 8 types of convergent subarrays and resulting 1-bit, -bit and 3-bit focusing-phase metasurfaces using the sequences shown in Figure. Based on these phase distributions, we can easily achieve the top and bottom meatasurface layouts by conducting a geometrical mapping process, see Figure S9. As can be seen, the layouts of the focusing-phase and uniform-phase metasurfaces are quite different, accounting for the completely different scattering S9

10 behavior of aforementioned metasurfaces discussed in the main text and below. Figure S8 Phase distributions of the (a) 8 convergent subarrays and (b) 1-bit, -bit and 3-bit coding metasurfaces, the 8 subarrays correspond orderly to the digits of ( or ), 1, 1 ( 1 ), 11, 1 ( 1 or 1 ), 11, 11 ( 11 ) and 111. S1

11 Figure S9 (a) Top and (b, c) bottom metallic layouts of the (a, b) focusing-phase and (c) already existed uniform-phase 1-bit, -bit and 3-bit coding metasurfaces of the same size. The 8 subarrays correspond orderly to the digits of ( or ), 1, 1 ( 1 ), 11, 1 ( 1 or 1 ), 11, 11 ( 11 ) and 111. Figure S1 portrays the FDTD calculated co-polarized and cross-polarized RCS reduction of the 1-bit focusing-phase metassurface as a function of frequency and reflection angle in E and H planes. As is shown, very similar scattering patterns are appreciated among three cases of x-polarized, y-polarized and RCP-polarized excitations, revealing a very robust scattering behavior independent on wave polarizations. In all cases, the waves scattered from the metasurface have been dispersed to S11

12 numberless angles across -9 o <θ<9 o for both co- and cross-polarized components. Such diffusion behavior, intrinsic and deterministic without optimizations, is insensitive to the global coding sequences (metasurface layout), which has been validated by extensive theoretical and FDTD calculations using other coding sequences. Although the maximum scattering of our focusing-phase metasurface mainly occurs around backscatter θ= o, both monostatic and bistatic RCS (Figures S1a-S1g) has been significantly reduced within 6.9~17.6 GHz with respect to the backscatter of a metallic plate (Figure S1h). Figure S1 FDTD calculated D bistatic RCS reduction of 1-bit focusing-phase metasurface in (a-f) E plane and (g) H plane. (a, c, e, g) Co-polarized and (b, d, f) cross-polarized RCS reduction for (a, b) x polarization, (c, d) y polarization and (e, f) RCP polarization. (h) The reflection distributions of the S1

13 same-sized metallic plate. Figure S11 further depicts the FDTD calculated co-polarized and cross-polarized RCS reduction of the -bit focusing-phase metassurface as a function of frequency and reflection angle in E plane. Again, almost the same scattering behavior as those shown in Figure S1 is observed for both x-polarized and LCP-polarized wave excitations. Therein, we conclude that diffusion behavior of our focusing-phase metasurfaces is insensitive to the coding bits. Figure S11 FDTD calculated D bistatic RCS reduction of the -bit focusing-phase metasurface in E plane. (a, c) Co-polarized and (b, d) cross-polarized RCS reduction under (a, b) x polarization and (c, d) LCP polarization. Figure S1 further depicts the consistent numerically and theoretically calculated scattering patterns of the -bit and 3-bit metasurfaces at other three representative frequencies. As is expected, the focusing-phase metasurface exhibits innumerable scattering beams, which are similar as those shown in Figure 7 in the main text. However, sharp scattering at specific angle is again expected in all cases for the already existed uniform-phase metasurface. Therefore, our approach enables diffusion with both decent monostatic and biastatic RCS reduction in broad operation band. S13

14 Figure S1 (a, c) Numerically and (b, d) theoretically calculated 3D scattering patterns of the (a, b) -bit and (c, d) 3-bit focusing- and uniform-phase metasurfaces at normal incidence. 7. FDTD results for a random-phase metasurface In this section, we afford the FDTD calculated results of the random-phase metasurface for further reference. As shown in Figure S13a, both the sequence and meta-atoms in subarrays were randomly generated within o φ 36 o without optimizations. Using the geometrical mapping approach, we can easily achieve the metasurface layout shown in Figure S13b. From the numerical results shown in Figures S13c-S13d, it is learned although the random-phase metasurface exhibits similar diffusion behavior as that of our focusing-phase one, the backscattering above 13 GHz is significantly larger than that of our focusing metasurface. Such enhanced backscattering of available approach reduces considerably the bandwidth which was divided from a broad band to several narrow bands. Therefore, time-consuming optimization is always necessary to achieve desirable broadband monostatic and bistatic RCS reduction for random-phase metasurface. S14

15 Figure S13 FDTD calculated diffusion behavior of the random-phase metasurface under LCP wave excitation. (a) Phase distribution. (b) Bottom-layer metallic layout. (c) Maximum and backscatter RCS reduction. (d) 3D scattering patterns. The results of 1-bit focusing-phase metasurface were afforded in c for comparison. 8. Additional experimental results for the focusing-phase metasurfaces In Figures 7, 8 and 9 of the main text, we show the measured scattering behavior of the 1-bit and -bit focusing-phase metasurface in H plane. Here, we further plot the measured reflection distributions of both metasurfaces in E plane under x-, y- and RCP-polarized excitations, see Figures S14 and S15. Similar diffusion behavior as that in H plane is observed in E plane for both samples, from which the scattering waves were dispersed toward enormous angles. However, the specular reflections dominate in E plane for the metallic plate. S15

16 Figure S14 Measured scattering behavior of the 1-bit focusing-phase metasurface. Co-polarized E-plane bistatic RCS reduction of (a, b) the metasurface and (c) same-sized metallic plate versus frequency and reflection angle under normal incidence, the reflection was recorded in an increment of 1 in the upper half-space with a dead zone of -5 <θ<5 shielded by the transmitting horn. (d) Monostatic RCS reduction. S16

17 Figure S15 Measured scattering behavior of the -bit focusing-phase metasurface under normal incidence. (a-c) Co-polarized E-plane bistatic RCS reduction versus frequency and reflection angle. (d) Monostatic RCS reduction. 9. Pictures of the experimental setup Figure S16 Illustration of the angle-resolved (a) monostatic and (b) bistatic reflection measurement setup. S17