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Abstract
Metasurfaces that can operate without a strictly periodic arrangement of meta-atoms are highly desirable for practical optical micro-nano devices. In this paper, we propose two kinds of Kerker-type metasurfaces that exhibit immunity to positional disorder. These metasurfaces consist of two distinct core-shell cylinders that satisfy the first and second Kerker conditions, respectively. Despite significant positional disorder perturbations of the meta-atoms, the metasurfaces can maintain excellent performance comparable to periodic ones, including total transmission and magnetic mirror responses. This positional disorder immunity arises from the unidirectional forward or backward scattering of a single core-shell cylinder, which results in minimal lateral scattering coupling between neighboring cylinders, thereby having little impact on multiple scattering in either the forward or backward direction. In contrast, the response of positional disorder non-Kerker-type metasurfaces decreases significantly. Our findings present a new approach for designing robust metasurfaces and expanding the applications of metasurfaces in sensing and communications within complex practical scenarios.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Two-dimensional (2D) planar metasurfaces with periodic sub-wavelength meta-atoms have gained significant interest in the field of efficient optical integrated components applied in communication [1], national defense [2], biomedicine [3], and other areas. Unlike classical optical path accumulation in continuous media, metasurfaces enable independent control of scattering amplitude, phase, and polarization of meta-atoms through the Pancharatnan-Berry phase, localized surface plasmon resonance, and Mie resonance [4], resulting in a minimal propagation path for light-matter interaction. Consequently, metasurfaces can precisely manipulate the electromagnetic (EM) wavefront at various frequencies by introducing abrupt changes in EM responses, and have also achieved remarkable advancements such as the novel generalized Snell law [5,6], photon spin Hall effect [7], planar microlens [8], vector beam [9], holography [10], etc.
Obtaining predetermined EM responses relies on arranging the meta-atoms, which serve as secondary sources, on the surface of a substrate in a specific global order, typically periodic [5,6,11–16]. However, within each unit cell, the azimuth, size, shape, and material of the meta-atom can vary randomly [17,18]. Achieving perfectly periodic subwavelength patterning structures necessitates sophisticated and precise fabrication technologies, such as photolithography, electron-beam lithography, and focused-ion-beam lithography, all of which are time-consuming and expensive [4,14,19]. In practical applications, the precision limitations of fabrication and external forces in unstable and harsh environments inevitably disrupt the orderly arrangement of the meta-atoms, failing to achieve the desired EM performances of the metasurfaces [20–23]. Furthermore, periodicity introduces significant spatial dispersion and limits the degree of manipulation freedom [24,25]. Hence, reducing the reliance on periodicity is preferable for metasurfaces to exhibit high-performance capabilities in the presence of positional disordered perturbations, leading to the development of disorder-immune metasurfaces.
Subsequently, when positional disorder perturbation is introduced, the EM responses of metasurfaces differ from those of periodic ones [26–28]. Consequently, a crucial aspect of achieving disorder-immune metasurfaces lies in minimizing the unpredictable couplings or interactions resulting from the disordered spacing between meta-atoms [29–31]. One effective approach to address this issue is to employ meta-atom that exhibit Mie resonances, which confine the EM field around or inside the particles, particularly when they are not in close proximity [32,33]. Another notable case is the non-radiative anapole mode, which confines the electric field inside the particles [34–36]. This mode can be applied to positional disorder immune metasurfaces due to the absence of coupling between particles [37]. However, resonant modes are not suitable for dealing with small spacing, and the non-radiative anapole mode lacks EM responses in the far field.
Encouragingly, particles exhibiting unidirectional scattering maintain this characteristic even in a random ensemble [38]. Kerker scattering, discovered by Kerker et al. during their study on the scattering of magnetic spheres, offers a promising solution [39]. Kerker scattering encompasses two types: the first (second) kind involves zero-backward (zero-forward) scattering, resulting from destructive interferences between the electric (ED) and magnetic (MD) dipoles in the backward (forward) direction, thus enabling forward (backward) directional scattering [39–42]. Subsequent research has further explored the generalized Kerker effect [43–45], lattice Kerker effect [46,47], acoustic Kerker effect [48], biological Kerker effect [49,50], optomechanical Kerker effect [51], and more. Another noteworthy advancement is the magnetic mirror, a high reflector that ensures an in-phase between the reflected and incident electric fields at the interface. This phenomenon enhances light-matter interactions and finds applications in subwavelength imaging [52], molecular fluorescence [53], Raman spectroscopy [54], perfect reflectors [55]. Additionally, Kerker-type metasurfaces offer the potential for antenna engineering [11,56] and reconfigurable intelligent surfaces [1,57], enabling low-loss and stable unidirectional signal transmission and reception, as well as achieving high-precision positioning.
In this paper, we proposed two types of positional disorder immune metasurfaces based on the Kerker scattering principle, each exhibiting distinct responses: total transmission and magnetic mirror effects. The first type of metasurface consists of hollow $\mathrm {Al_2O_3}$ cylinders that support the first kind of Kerker scattering, while the second type comprises copper-$\mathrm {Al_2O_3}$ cylinders that demonstrate the second kind of Kerker scattering. The presence of positional disorder has minimal impact on multiple scatterings of the meta-atoms due to their very weak interactions of the lateral scattering. As a result, the random Kerker-type metasurfaces maintain their total transmission and magnetic mirror responses even in the presence of significant positional disorder. In contrast, the performance of the random non-Kerker-type metasurface is significantly reduced due to unpredictable lateral scattering couplings. These lateral scatterings are primarily associated with the magnetic quadrupole (MQ) mode. The Kerker-type metasurfaces designed to be immune to positional disorder operate at a frequency of 10 GHz within the X-band range (8-12 GHz) [58]. The compact and lightweight nature of the devices within this frequency band makes them suitable for applications such as satellite communications, RAdio Detecting And Ranging (RADAR), monitoring small-scale weather systems and their evolution [59], and high-precision radiotherapy [60]. Moreover, our finding is also suited for unidirectional scattering in other angles beyond the forward and backward directions. Consequently, the Kerker-type metasurfaces exhibit high robustness, offer the potential for single-step fabrication, and enhance the adaptability of metasurfaces in various practice environments for applications in sensing, wireless communications, and photonics.
2. Results and discussions
2.1 Disorder immunity of the first kind of Kerker metasurface
We begin by providing a meta-atom composed of an infinite-long hollow cylinder in Fig. 1(a). The inner and outer radii of the cylinder are $r_0$ and $r_1$, respectively. The incident plane electromagnetic (EM) wave is polarized with the electric field $\mathbf {E}$ parallel to the axis of the cylinder and the incident frequency is 10 GHz. The shell is made of $\mathrm {Al_2O_3}$ with relative permittivity $\varepsilon _1=9.424+i2.920 \times 10^{-3}$ [61] at the specified frequency. Here, we set the size parameter $x_r= {k}r_1$ as 1.343, where k represents the wavenumber of the incident EM wave in the air. For simplicity, we introduce the dimensionless parameter $\alpha =r_0/r_1$.
Fig. 1. The first kind of Kerker-type metasurface. (a) Schematic of scattering analysis of an individual cylinder. (b) Scattering efficiencies of a single cylinder varying $\alpha$. The red curve (sca) is the total scattering. Point A denotes $\alpha =0.185$. (c) Angular scattering of point A. (d) Schematic of periodic metasurface with minimal face spacing p under the normally incident plane wave with wavevector ${\mathbf {k}}$. (e) Transmissivity (T) responses of the metasurfaces concerning $\alpha$. Each curve has a fixed p, which varies from 0.05$\lambda$ to 0.45$\lambda$ with an interval of 0.05$\lambda$. (f) Reflectivity (R) and T responses of the finite-size metasurfaces at point A. Rp (Tp) is the R (T) of the periodic metasurfaces. Rr (Tr) is the R (T) of the random metasurfaces. The mR (mT) is the average value of all Rr (Tr).
Then, we analyze the scattering properties of an individual cylinder. The semi-analytical scattering multipole decomposition, employing Cartesian bases, is based on displacement current density, which is expressed as [62]
Subsequently, the total multipole moment $(\mathbf {D}+i{k}\mathbf {V})$ is calculated by the interference between the base Cartesian moment $\mathbf {D}$ (representing the arbitrary Cartesian multipole moment in $\mathbf {D}^{e}$, $\mathbf {D}^{m}$, and $\mathbf {D}^{mq}$) and its toroidal moment $\mathbf {V}$. The scattering contribution of a multipole is proportional to $|\mathbf {D}+i{k}\mathbf {V}|^{2}$ [64].
Based on Eqs. (1)–(7), the scattering efficiencies and the summation of all modes are shown in Fig. 1(b). At point A ($\alpha =0.185$), the scattering efficiencies of ED and MD are equal, and the MQ is very weak. According to the spectral region of point A, the induced ED and MD modes are in phase, effectively suppressing radiation in the backward direction [41,42]. Additionally, the analysis of the anapole mode excited by the cylinder can be seen in Fig. S1(a) in Supplement 1.
Considering the induced multipole moments of the ED, MD, and MQ, the angular scattering intensity reads as [64,68–70]
Fig. 2. Frequency spectra of the first kind of Kerker-type metasurface. (a) Transmissivity and Reflectivity responses of infinite periodic metasurface (T and R), finite periodic metasurface (Tp and Rp), and the average values of 10 random metasurfaces (mT and mR), respectively. The error bar denotes the standard deviation of random samples. (b) Scattering electric field ($|\mathbf {E}_\mathrm {s}|$) distribution of the cylinder at point A. The scale bar is 5 mm.
We propose a metasurface composed of periodically arranged core-shell cylinders along the $x$-direction with the minimal face spacing $p$ depicted in Fig. 1(d). The metasurface is normally illuminated by a plane EM wave with a $z$-polarized $\mathbf {E}$. In our study, we utilize the commercial software COMSOL [73]. The periodic boundary condition is applied along the $x$-direction and the perfectly matched layer (PML) is used to replace the outermost air along the $y$-direction. When $p$ varies from 0.05$\lambda$ to 0.45$\lambda$ with an interval of 0.05$\lambda$, Fig. 1(e) depicts the transmissivity (T) spectra of the metasurfaces as functions of $\alpha$ and $p$. Notably, the transmissivity spectra exhibit peaks with identical values (about 0.99) near point A. Therefore, the metasurface maintains total transmission across a wide range of $p$ at point A, where the first kind of Kerker condition is approximately satisfied.
The periodic metasurface is truncated to only 51 cylinders with $p=0.25\lambda$, and the reflectivity (Rp) and transmissivity (Tp) of the finite-size periodic metasurface at point A are illustrated in Fig. 1(f). To simulate the finite-size structures, PMLs are implemented in all directions. The metasurface still exhibits the nearly total transmission with Tp being about 0.99 and $\mathrm {Rp}=0$. Then, the positions of cylinders are perturbed with $p$ varying from 0.05$\lambda$ to 0.45$\lambda$ randomly. The disorder degree ($\sigma$) of the metasurface is defined as $\sigma =\mathrm {max}(|\Delta p_{i}/p_{0}|)$, where $\Delta p_{i}= p_{i}-p_{0}$ with $p_i$ being the $p$ of arbitrary adjacent cylinders and $p_{0}$ being the initial $p$. Ten different random metasurfaces with $\sigma =80{\%}$ are prepared for analysis. The reflectivity (Rr) and transmissivity (Tr) are measured for these random samples. The average value of all Tr is mT about 0.94 decreasing only 0.05 compared to the Tp, while the average value of all Rr is mR still zero. Notably, the red dots are located on the red line with a deviation of about 0.0055. Therefore, the transmission of random metasurfaces exhibits strong stability.
The frequency spectra of the Kerker-type metasurfaces are depicted in Fig. 2(a). The R and T of the infinite periodic metasurface indicate the response of the proposed structure is highly dependent on the incident frequency, with the total transmission occurring only at 10 GHz. The Rp and Tp of finite-size periodic metasurface closely match the R and T, respectively, suggesting minimal influence from the structure width. However, the mR and mT of the random metasurfaces at frequencies far from 10 GHz deviate and fluctuate significantly compared with Rp and Tp. In contrast, the deviation is small and the response becomes more stable near 10 GHz, with the smallest deviation observed at this frequency. The variation of frequency of the fixed structures results in changes in scattering modes that no longer satisfy the Kerker scattering condition, leading to a decrease in the positional disorder immunity. Therefore, the proposed positional disorder immune Kerker metasurface exhibits a certain frequency bandwidth with the best positional disorder immunity observed at 10 GHz.
Figure 3(a) presents the far-field radiations of the finite-size periodic (per) and random (ran) metasurfaces. The primary EM responses in both cases are the zeroth-order transmission and the two peaks exhibiting almost identical characteristics. This similarity arises from the negligible radiation in other angular directions. Figures 3(b) and 3(c) depict the $\mathbf {E}$ distributions of the periodic and random metasurfaces, respectively. The transmitted wavefront is perturbed weakly due to the appearance of weak radiations in the non-zero-degree directions, notwithstanding the random metasurface still maintains almost total transmission.
Fig. 3. Responses comparisons of the periodic (per) and random (ran) metasurfaces at point A. (a) The far-field $|\mathbf {E}|$ of the per and ran metasurfaces. (b) and (c) The $\mathbf {E}$ distributions of the per and ran metasurfaces, respectively.
Next, we examine the multiple scatterings of the cylinders at point A. Figure 4(a) shows the angular scattering patterns of two cylinders as $p$ varies from 0.05$\lambda$ to 0.45$\lambda$. The lateral scatterings of all curves are very weak. Despite the weak coupling slightly decreasing the magnitude of peaks when the particles are very close, the curve shape of the unidirectional forward scattering of the two cylinders is robust against the variation of $p$. Figure 4(b) depicts the multiple scatterings of N cylinders with $p=0.25\lambda$. All patterns show the unidirectional forward scattering, accompanied by effective suppression of backward and lateral scatterings. Furthermore, the magnitude of the peak approximately scales with N, satisfying the principles of grating diffraction [74]. Therefore, the disorder immunity of the metasurfaces originates from the weak lateral scattering of the cylinders with exciting the first kind of Kerker scattering, which leads to the negligible lateral scattering coupling that rarely affects the multiple scatterings.
Fig. 4. Multiple scatterings of the cylinders at point A. (a) Angular scattering patterns of the two cylinders varying $p$ from 0.05$\lambda$ to 0.45$\lambda$. (b) Angular scattering of N cylinders with fixed $p=0.25\lambda$. N denotes the number of cylinders increasing from 1 to 7.
In practical applications, it is also important to consider the impact of the disordered perturbations on the size and shape of the cylinders. Compared to the larger outer diameter of a cylinder, errors during the manufacturing process are more likely to occur in the smaller inner diameter. Therefore, our focus is primarily on investigating size and shape perturbations related to the inner radius. Here, the positions of the cylinders follow a periodic arrangement. Figure 5(a) illustrates the reflectivity and transmissivity of the finite-size Kerker metasurfaces with disordered size perturbation in the inner radii. The size disorder degree is defined as $\sigma _{r}=\mathrm {max}(|\Delta r_{0i}/r_{0}|)$, where $\Delta r_{0i}= r_{0i}-r_{0}$ with $r_{0i}$ being the $r_{0}$ of arbitrary cylinder. We prepared 10 different random Kerker metasurfaces with $\sigma _{r}=40{\%}$. The mT is about 0.88 decreasing by 0.11 compared to the Tp, while the mR remains near to zero. The positions of the red dots exhibit significant oscillations with a standard deviation of approximately 0.016.
Fig. 5. Transmissivity and Reflectivity responses of the disordered size and shape perturbations in the cylinders of the first kind of Kerker-type metasurface. (a) Inner radii perturbations of cylinders with a disorder degree of $40{\%}$. Transmissivity (Tr) and Reflectivity (Rr) of the Kerker metasurface with disordered size perturbations of the cylinders. (b) Shape perturbations of air elliptical cylinders with a disorder degree of $60{\%}$. Tr and Rr of the Kerker metasurface with disordered shape perturbations of the cylinders. Inset is the schematic of a cross-section of an elliptical cylinder achieved by changing the inner radius of the cylinder in the $x$-direction.
On the other hand, we fix the inner radius in the $y$-direction ($r_{0y}$) while only varying the inner radius in the $x$-direction ($r_{0x}$) to create different air elliptical cylinders, as shown in the inset of Fig. 5(b). The disorder degree of shape perturbation random metasurface is defined as $\sigma _{s}=\mathrm {max}(|\Delta r_{0xi}/r_{0}|)$, where $\Delta r_{0xi}= r_{0xi}-r_{0}$ with $r_{0xi}$ being the $r_{0x}$ of arbitrary cylinder. In Fig. 5(b), with the $\sigma _{s}=60{\%}$, the mT is 0.91 decreasing by 0.08 compared to the Tp, while the mR is about zero. The standard deviation of the Tr is about 0.012. Consequently, compared to the positional disorder Kerker metasurfaces, these metasurfaces with smaller disorder degrees of size and shape exhibit significantly weaker disorder immunities. The underlying reason is that the change in inner diameter size can alter parameter $\alpha$, resulting in the scattering mode of the cylinder not meeting the Kerker scattering condition. While the change in inner diameter is somewhat equivalent to the change in outer diameter, the latter introduces greater surface relief, resulting in increased diffuse reflection, which further diminishes the dominant responses and stability. Detailed discussions on various shape perturbations are beyond the scope of this paper.
2.2 Disorder immunity of the second kind of Kerker metasurface
Now, we turn our attention to the second kind of Kerker metasurface. In Fig. 6(a), the meta-atom is an infinitely long copper-$\mathrm {Al_2O_3}$ core-shell cylinder, and the perfect electric conductor (PEC) can displace the copper at the incident frequency. Under the plane EM wave, Fig. 6(b) shows the scattering efficiencies of total scattering (sca), ED, and MD modes of the isolated cylinder, where $x_r$ is 0.785. The other weak higher-order modes are neglected. Point B denotes $\alpha =0.432$ and the equal contributions of ED and MD. According to the spectral region of point B, the induced ED and MD modes are anti-phase leading to scattering cancellation in the forward direction [42].
Fig. 6. The second kind of Kerker-type metasurface. (a) Schematic of scattering analysis of an individual cylinder. (b) Scattering efficiencies of a single cylinder. Point B denotes $\alpha =0.432$. (c) Angular scattering of the cylinder at point B. (d) Schematic of the periodic metasurface. (e) The phase difference ($\Delta \phi _\mathrm {r}$) of the reflected and incident electric field on the top interface plane of the metasurface when $x_r=0.785$ as $\alpha$ varies. Each curve has a fixed $p$ changing from $p_0$ (0.05$\lambda$) to $p_1$ (0.45$\lambda$) with a step of 0.05$\lambda$. (f) Reflectivity and $\Delta \phi _\mathrm {r}$ responses of the finite-size metasurfaces at point B. Pp: the $\Delta \phi _\mathrm {r}$ of the periodic metasurface; Pr: the $\Delta \phi _\mathrm {r}$ of the random metasurface; mP: the average value of all Pr.
The angular scattering of the individual cylinder at point B is depicted in Fig. 6(c). At this point, the interference of ED and MD results in strong total scattering (sca) in the backward direction, whereas the forward and lateral scatterings nearly vanish. The incident wave and the nonzero imaginary part of $\varepsilon _1$ hinder complete suppression of the forward scattering, while forward scattering can disappear in the gain material [40]. Nonetheless, the backward-to-forward scattering ratio is as high as 11.25, and $g=-0.60$. Here, $\beta =4.87$, which indicates the lateral scattering remains weak. Additionally, Fig. 7(b) shows the $|\mathbf {E}_\mathrm {s}|$ distribution of a single cylinder at point B. The near-field distribution of backward unidirectional scattering fits well with the theory in Fig. 6(c). Therefore, the cylinder excites the unidirectional backward scattering satisfying the second kind of Kerker scattering nearly [45]. Moreover, the analysis of the anapole mode in the cylinder is presented in Fig. S1(b) in Supplement 1.
Fig. 7. Frequency spectra of the second kind of Kerker-type metasurface. (a) R and $\Delta \phi _\mathrm {r}$ responses of infinite periodic metasurface (R and P), finite periodic metasurface (Rp and Pp), and the average values of 10 random metasurfaces (mR and mP), respectively. The red and blue error bars of the random samples have been magnified 10 and 15 times, respectively. (b) $|\mathbf {E}_\mathrm {s}|$ distribution of the cylinder at point B. The scale bar is 5 mm.
The schematic diagram of a total reflective periodic metasurface composed of cylinders is shown in Fig. 6(d). The metal substrate acting as an electric mirror ensures total reflection and a phase difference of $\pi$ [see Fig. S2 in Supplement 1]. Figure 6(e) demonstrates the phase difference ($\Delta \phi _\mathrm {r}$) between the reflected and incident electric field at the top tangent plane of the metasurface as $p$ varying from $p_0$ (0.05$\lambda$) to $p_1$ (0.45$\lambda$) in a step of 0.05$\lambda$. All curves change continuously from negative to positive values and intersect at point B where $\Delta \phi _\mathrm {r}=0$. Therefore, the metasurface at point B is a magnetic mirror over a broad range of $p$.
The EM responses of finite-size periodic and random metasurfaces at point B are illustrated in Fig. 6(f). The reflectivity (Rp) and $\Delta \phi _\mathrm {r}$ (Pp) of the periodic metasurface with $p=0.25\lambda$ are 0.99 and $-0.0091$ rad, respectively. Thus, the finite-size periodic metasurface still exhibits the magnetic mirror response. Then, we prepare 10 random metasurfaces when $\sigma =80{\%}$ and $p_i$ is randomly in the range from 0.05$\lambda$ to 0.45$\lambda$. The Pr is the $\Delta \phi _\mathrm {r}$ of the random metasurface and the mP is the average of all Pr. In this case, $\mathrm {mR}=0.98$ and $\mathrm {mP}=-0.0089$ rad. The Rp, Rr, and mR have almost identical high reflectivity, and the $\Delta \phi _\mathrm {r}$ of blue data remains about 0. As a result, the random metasurfaces still exhibit the magnetic mirror response.
Figure 7(a) shows the frequency spectra of the Kerker-type metasurfaces. The infinite periodic metasurface retains the nearly total reflection, while the $\Delta \phi _\mathrm {r}$ is sensitive to the incident frequency. The responses of finite-size random Kerker metasurfaces are nearly consistent with finite-size periodic metasurfaces and the optimal immune effect appears at 10 GHz. Conversely, the mR and mP deviate and fluctuate more significantly at frequencies distant from 10 GHz. Notably, the red and blue error bars have been magnified 10 and 15 times, respectively. Therefore, the magnetic mirror response of the Kerker-type metasurface demonstrates the positional disorder immunity within the specific frequency range.
The far-field radiations of the finite-size periodic (per) and random (ran) metasurfaces at point B are compared in Fig. 8(a). The two curves perfectly overlap indicating the random and periodic metasurfaces excite the same zeroth-order total reflection. Figure 8(b) shows the $|\mathbf {E}|$ distribution of the periodic metasurface at point B, and Fig. 8(c) is the case of a random one. All metasurfaces exhibit the magnetic mirror response with a large-scale strongest interference electric field at their top interfaces. The wavefront perturbations primarily arise from the diffuse reflection at the side boundaries, while in the case of the random metasurface also includes the interference of weak non-zeroth order diffractions. Therefore, the magnetic mirror response of the second kind of Kerker metasurface is robust to large disordered position perturbation.
Fig. 8. (a) The far-field $|\mathbf {E}|$ of the periodic and random metasurfaces at point B. (b) and (c) The fragments of $|\mathbf {E}|$ distributions of the periodic and random metasurfaces at point B, respectively. (d) Intersection point (see point B in Fig. 6(e)) of the $\Delta \phi _\mathrm {r}$ as a function of $x_r$. The acquisition method is as same as that described in Fig. 6(e). The red line is the fitting of all data. (e) Schematic of the effective structure of the metasurface with the same incident wave, the thickness equals the external diameter of the cylinder.
We control the $\Delta \phi _\mathrm {r}$ of the phase curves’ intersection point of the infinitely periodic metasurface as a function of $x_r$ varying $p$ from 0.05$\lambda$ to 0.45$\lambda$ in Fig. 8(d). The underlying physical mechanisms of all points are the same as point B. However, the point will disappear due to the exciting mode change when $x_r$ exceeds out of the range in this figure. The points are fitted by a line with a slope of $-1.27$. As $x_r$ increases, $\Delta \phi _\mathrm {r}$ of the intersection point decreases linearly, with point B lying precisely on the line. In Fig. 8(e), the metasurface can be viewed as a PEC substrate and a dielectric film with a height of $2r_1$. Under the normal incidence of a plane wave, the $\Delta \phi _\mathrm {r}$ at the top interface of the structure reads as $\Delta \phi _\mathrm {r}=4kn_{eff}r_1$, where $n_{eff}$ is the effective refractive index. According to the slope, the $n_{eff}$ is 0.995. Therefore, the cylinders in the metasurface are equivalent to an air-like film for any non-invisible passive systems, and the $\Delta \phi _\mathrm {r}$ can be linearly adjusted by altering the thickness. This linear control over the reflected phase of the metasurface achieved by manipulating $x_r$ while $p$ changes within a wide range is beneficial for applications such as positional disorder immune gradient metasurfaces to achieve the reflection deflection, carpet cloak, etc [5,37].
Figure 9(a) illustrates the multiple scatterings of two cylinders at point B as the $p$ varies from 0.05$\lambda$ to 0.45$\lambda$. All patterns display strong backward scattering and suppressed forward and lateral scatterings. The main lobe beamwidth and magnitude decrease at a large $p$, which indicates the weak coupling vanishes. Figure 9(b) depicts the multiple scatterings of N cylinders at point B with $p=0.25\lambda$. Consequently, the cylinders still exhibit high unidirectional backward scattering. Therefore, the disorder immunity of the metasurface arises from the weak lateral scattering of the cylinder with exciting the second kind of Kerker scattering, which leads to very weak lateral scattering coupling.
Fig. 9. Multiple scatterings of the cylinders at point B. (a) Angular scatterings of two cylinders as $p$ varying. (b) Angular scatterings of N cylinders with fixed $p=0.25\lambda$.
We also investigate the effects of disordered perturbations in the size and shape of cylinders when their positions are arranged periodically. Figure 10(a) presents the Rr and Pr of 10 finite-size Kerker metasurfaces with disordered inner radii size perturbations. The disorder degree $\sigma _r$ is $40{\%}$. The $\mathrm {mR}=0.26$ and $\mathrm {mP}=-0.18$ rad, which are significantly lower than the values of the positional disorder Kerker metasurfaces, respectively. Moreover, the standard deviations of the Rr and Pr are 0.071 and 0.21 rad, respectively, both larger than the values of the positional disorder metasurfaces. In the same way, Fig. 10(b) depicts the responses of finite-size Kerker metasurfaces with disordered shape perturbations of the elliptical cylinders. In this case, the disorder degree $\sigma _s$ is $60{\%}$. The $\mathrm {mR}=0.51$ and $\mathrm {mP}=-0.11$ rad. The standard deviations of the Rr and Pr are 0.058 and 0.21 rad, respectively. The significantly decreased size and shape disorder immunities indicate the responses of the metasurfaces are more sensitive to the size changes in the cylinder due to the size change leading to the mismatch of the Kerker scattering condition.
Fig. 10. Reflectivity and $\Delta \phi _\mathrm {r}$ responses of the disordered size and shape perturbations in the cylinders of the second kind of Kerker-type metasurface. (a) Inner radii perturbations of cylinders with a disorder degree of $40{\%}$. Rr and $\Delta \phi _\mathrm {r}$ (Pr) of the Kerker metasurface with disordered size perturbations of the cylinders. (b) Shape perturbations of elliptical cylinders with a disorder degree of $60{\%}$. Rr and Pr of the Kerker metasurface with disordered shape perturbations of the cylinders. The elliptical cylinder is achieved in the same way.
2.3 Disorder immunity of the non-Kerker-type metasurface
In this section, we focus on the non-Kerker-type metasurface case. Figure 11(a) shows the meta-atom is an $\mathrm {Al_2O_3}$ cylinder. Under the same incident wave, Fig. 11(b) demonstrates the scattering efficiencies of the total scattering (sca), ED, MD, and MQ modes. These modes have equal contributions at point C with $x_r=1.15$. In Fig. 11(c), the angular scattering of the sca shows the non-unidirectional scatterings of the cylinder at point C. Here, $g=0.41$ indicates more scattering power in the forward direction. $\beta =3.66$ lower than the Kerker metasurfaces values, indicating the lateral scattering becomes stronger. The near-filed scattering electric field $|\mathbf {E}_\mathrm {s}|$ distribution of the cylinder at point C proves the theoretical results [see Fig. S3 in Supplement 1].
Fig. 11. One kind of non-Kerker-type metasurface. (a) Schematic of scattering analysis of an individual cylinder with permittivity $\varepsilon _1$. (b) Scattering efficiencies of a single cylinder. Point C denotes $x_r=1.15$. (c) Angular scattering pattern of the cylinder at point C. (d) Schematic of the periodic metasurface. (e) The R responses of the metasurfaces concerning $x_r$. Each curve has a fixed $p$ and the arrow denotes the $p$ increasing from 0.25$\lambda$ to 0.51$\lambda$. (f) R and T responses of the finite-size periodic and random metasurfaces at point C.
Figure 11(d) demonstrates the schematic of the periodic metasurfaces. The reflectivities of the metasurface as the functions of $x_r$ and $p$ are shown in Fig. 11(e), where $p$ varies from 0.25$\lambda$ to 0.51$\lambda$ with a step of 0.01$\lambda$ and the arrow denotes the increasing direction. Interestingly, all curves nearly overlap at point C with $\mathrm {R}>0.95$. Therefore, the metasurfaces at point C have almost the same high reflectivities within that range of $p$.
Figure 11(f) depicts the reflectivity and transmissivity of the finite-size metasurfaces at point C. The periodic metasurface with $p=0.38\lambda$ exhibits $\mathrm {Rp}=0.98$ and $\mathrm {Tp}=0$. Then, 10 random metasurfaces with $\sigma =34.21{\%}$ are achieved by disordered distributing each $p$ of the periodic metasurface in the range of 0.25$\lambda$ to 0.51$\lambda$. The distributions of Rr and Tr indicate the random metasurfaces are still high reflection without transmission. However, the $\mathrm {mR}=0.82$ decreased by 0.16 compared with the Rp, and the deviation of all Rr is about 0.014. Therefore, compared to the periodic metasurface, the response of the random metasurface falls off and fluctuates significantly.
The far-field radiations of the finite-size periodic and random metasurfaces are shown in Fig. 12(a). Both show dominant zeroth-order reflection but the random metasurface shows more pronounced radiations in other directions. Hence, the decrease of the Rr is mainly ascribed to the increased radiations in other directions. The $|\mathbf {E}|$ distribution of the periodic metasurface in Fig. 12(b) indicates the high reflection originates from the waveguide-array (WGA) modes [75] and the lateral coupling between adjacent cylinders. Then, the $|\mathbf {E}|$ distribution of the random metasurfaces in Fig. 12(c) demonstrates the lateral couplings of the adjacent cylinders still occur but their strengths are different. Furthermore, the radiations of non-backward directions lead to the reflected wavefront perturbation and the speckles below the random metasurface.
Fig. 12. Response comparisons of the periodic and random metasurfaces at point C. (a) The far-field $|\mathbf {E}|$ of the per and ran metasurfaces. (b) and (c) electric field $|\mathbf {E}|$ distributions of the per and ran metasurfaces, respectively.
We explore the multiple scatterings of two cylinders at point C in Fig. 13(a). The change of $p$ not only changes the magnitude and main lobe width of the forward and backward scatterings but also produces different lateral scatterings, thereby influencing the lateral scattering couplings between two cylinders. The various lateral couplings cause the non-backward radiations of the random metasurfaces and lead to the energy loss of the zeroth-order reflection and increased instability. Figure 13(b) demonstrates the multiple scatterings of N cylinders with $p=0.38\lambda$. The scatterings maintain the dominant forward and backward scatterings. Therefore, the non-Kerker metasurface has a lower disorder immune degree. Moreover, the investigation of size and shape perturbations of the non-Kerker metasurfaces is shown in Fig. S4 in Supplement 1.
Fig. 13. Multiple scatterings of the cylinders at point C. (a) Angular scattering patterns of the two cylinders varying $p$ from 0.25$\lambda$ to 0.49$\lambda$. (b) Angular scattering of N cylinders with fixed $p=0.38\lambda$.
Based on Eqs. (1)–(7), the induced displacement current is solely present in the dielectric layer and the scattering multipoles are influenced by the effective width ($w_{eff}=2(r_{1}-r_{0})$) of the dielectric layer. We suppose the parameter $\eta$ is the ratio of the effective width to the incident wavelength. Thus, the $\eta$ of the second kind of Kerker metasurface, the first kind of Kerker metasurface, and the non-Kerker metasurface are 0.14, 0.35, and 0.37, respectively. Considering the multipole analysis, we observe that the intensity of the MQ mode increases proportionally to $\eta$ or the effective width. Our findings indicate that a stronger MQ mode participates in the interference with ED and MD modes, resulting in a mismatch of Kerker scattering conditions and significant lateral scattering. As a result, the positional disorder immunity of the metasurface decrease. Hence, in our proposed cylinders, the effective width should be taken into account to mitigate the influence of higher-order MQ modes. However, this does not imply that higher-order modes cannot be employed in positional disorder immune metasurfaces. For instance, the interferences of ED, MD, MQ, and EQ modes can generate high directional scattering without inducing lateral scattering, known as generalized Kerker scattering [43].
3. Conclusions
In summary, we have proposed two kinds of Kerker-type metasurfaces exhibiting total transmission and magnetic mirror responses immune to the large positional disorder perturbations of the meta-atoms. The meta-atoms of the two Kerker-type metasurfaces satisfy the first and second kinds of Kerker conditions, respectively. Because of the vanishing lateral scattering, the lateral couplings between meta-atoms are weak. As a result, the dominant EM responses of Kerker-type metasurfaces are highly robust against large positional disorder perturbations. In contrast, for the non-Kerker-type metasurface, the positional disorder will affect the non-ignorable lateral coupling between meta-atoms and thus decrease the dominant EM responses. In our proposed cylinders, increasing the effective width of the dielectric layer enhances the MQ mode leading to stronger lateral scattering. Additionally, Kerker-type metasurfaces are very sensitive to changes in the size of the cylinders due to the mismatch of the Kerker scattering conditions.
This mechanism can be applied to design other positional disorder immune metasurfaces. In addition, the high unidirectional scattering is not limited to the forward and backward directions of ED and MD modes and also can be extended to other directions by optimizing high-order electric and magnetic modes, e.g. achieving the generalized Kerker scattering [43]. The Kerker-type positional disorder immune metasurfaces are valid across a wide frequency range in the X-band, making them suitable for satellite communications in complex cosmic environments that involve large temperature differences and space debris. Moreover, our findings can be scaled to other frequency ranges due to the scaling properties of the Maxwell Equations [76]. This work paves a new way for designing highly robust metasurfaces and sheds new light on the practically accessible metasurfaces that suit complex circumstances. In future work, we plan to explore the Kerker-type metasurfaces that are immune to incident polarization and angle by breaking the rotational symmetry of the meta-atoms [77].
Funding
Key project of National Key Research and Development Program of China (2022YFA1404500); National Natural Science Foundation of China (12074267); Basic and Applied Basic Research Foundation of Guangdong Province (2020A1515111037); Shenzhen Fundamental Research Program (20200814113625003).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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