1. Introduction

The recent emergence and rapid developments of light scattering by an individual meta-atom and metasurfaces have led to a wide range of advanced functionalities, such as optical cloaking [1], negative refractive index [2], beam splitters [3], holograms [4, 5], vortex masks [6, 7] and so on. Compared with metallic counterparts, dielectric materials [8], for example Si, Ge and GaAs, exhibit significantly lower losses in the visible and infrared regions with the possibilities to efficiently excite high-order Mie modes. Appropriate excitations and interferences between these modes pave a new paradigm for the design of metadevices with high efficiency and tunable directional functionalities, such as perfect reflectors and Huygen’s metasurfaces [9, 10], high-efficiency optical polarizers [11], etc.

In terms of directional scattering of an individual particle, the investigation was originally proposed by Kerker et al. in 1983 [12]. That is, under certain conditions for the permittivity and permeability of the particle, the optical response to plane-wave illumination consists of equal amplitude for electric dipoles (EDs) and magnetic dipoles (MDs). Though it has been well known for a long time, perfect forward scattering (FS) at visible [13] or near-infrared [14] regimes have not been experimentally demonstrated until recently, thanks to the excitations of strong magnetic response in nonmagnetic particles via artificial magnetism created by subwavelength structures [15]. Then the strong FS or backward scattering (BS) patterns can be well shaped based on instructive or destructive interferences between EDs and MDs. Inspired by the pioneering works [13–15], tailoring directional scattering through meta-dielectric or all-dielectric nanoparticles have attracted much attention [16–21]. While the Kerker’s conditions were initially derived for spherical particles [12], it has been demonstrated that the principles can be generalized to more complex geometries such as cylinders [20, 22], nanodisks [23] or nanopillars [24], etc. Besides, higher orders of Mie resonances can also be properly excited and tuned to meet the optimal FS condition, which has recently been achieved with a metal ring in THz frequencies depending on manipulating the amplitude and phase of the ED and electric quadrupole (EQ) [25]. Furthermore, controlling over the directional scattering can also be obtained using arrangements of elements like dimers [26], trimers or nanoparticles clusters [27] and even anisotropic nanostructures [20].

Although meta-atoms with desirable scattering directionality have been widely investigated, there is still something missing. Firstly, up to now achieving optimal BS is seldom discussed due to the difficulty in realizing zero-FS [28, 29] and low BS efficiency [18]. Actually, realizing perfect BS is of fundamental and practical importance in various novel applications, including optical cloaking [1], optical metacages [30], perfect reflectors [31], high-efficient polarizers [32] and so on. Secondly, despite of the pioneering work experimentally demonstrating the strong FS using dielectric materials, a number of later theoretical studies, on the contrast, paid much attention to fictitious materials with wavelength-independent optical constants [19, 21, 23]. Though the results may be instructive to some extent, the demonstrations by utilizing realistic materials with consideration of intrinsic absorption [33] are still relatively rare. Besides, the complex structures such as thin nanorings [25] are consequently more demanding for advanced fabrication technologies, making them difficult to fabricate. As a result, shaping the directional radiation patterns, especially for achieving strong BS using much simpler subwavlength nanoparticles with realistic materials is of great necessity and importance.

On the other hand, stimulated by the proposal of the generalized laws of reflection and refraction [34, 35], we have seen vibrant and flourishing developments in metasurfaces [36, 37] which have offered a completely new way to control light-matter interaction (including phase, amplitude and polarization of electromagnetic waves) in sub-wavelength scale [38–43]. Especially for Huygens’ metasurface based on Huygens’ sources (via the first Kerker’s condition) and Huygens’ reflectors (via the second Kerker’s condition), they offer a new way to realize full-wave control with high efficiency owing to resonant excitations and interferences of Mie modes [44–46]. For example, ideally 100% transmission efficiency and almost arbitrary spatial distributions of phase discontinuities have been theoretically and experimentally achieved in the near-infrared regime using arrays of silicon nanodisks [47]. Nevertheless, the role of lattice couplings [48] ascribed to particle arrangements in multipolar interferences [49] has not been discussed in detail, since the existing work mainly focused on the phenomenological performance or collective functionalities of the metasurfaces [43]. Studying lattice couplings is of great importance to unveil the physical mechanism behind novel optical phenomena. Specifically, the optical responses taking advantages of multipolar interferences and lattice couplings, can be very different from those of an isolated scatterer, providing an extra degree of freedom to design novel metadevices with improved efficiency. For example, by combining the effects of diffraction and multipolar interferences, it is achievable to steer light in certain directions with desired diffraction orders and unitary efficiency [50]. Besides, large-angle beam steering, perfect reflection and perfect transmission can also be realized when employing coherent lattice couplings to manipulate the excitation efficiencies of the optical modes [51]. Therefore, it is worth exploring the connections between them systematically.

Here, firstly, for an individual particle, we propose a principle to study the directional scattering of a cylinder based on the scattering asymmetry factor g. We show that strong BS for one polarization and strong FS for the other polarization can be obtained using realistic material (c-Si) with experimental data. Stimulated by the polarization-dependent phenomena, we design a one-dimensional metalattice to achieve a linear polarizer by combining the lattice effects and dipolar interferences. We find that lattice effects induced by the typical configuration are important in enhancing dipolar excitation efficiency, with the possibility to make the perfect metalattice-based polarizer come ture. Furthermore, the interplays between lattice couplings and dipolar interferences have been unequivocally unveiled to explain the reason for the perfect performance. Finally, we further investigate the performance of our proposed linear polarizers when considering the lattice perturbations, the effects of the system sizes, different incident angles and different working wavelengths as well.

2. Directional scattering of an individual meta-atom

2.1. Theory: Directional manipulation based on the scattering asymmetry factor g

We start by considering the directional far-field scattering patterns of an infinite dielectric cylinder with a normally incident plane wave from free space. The incident light can be s-polarized with electric field along the axis of the cylinder or p-polarized with magnetic field parallel to the axis of the cylinder shown in Fig. 1(a). The scattering field can be expanded into a set of cylindrical harmonics with the Mie scattering coefficients $a_m^{s,p}$ for s-polarization and p-polarization, respectively, in which m denotes the cylindrical harmonic number. With the scattering coefficients in the hands, the scattering characteristics in near field and far field can be obtained easily. Meanwhile, the scattering efficiency can also be directly calculated by [52, 53]

 

Fig. 1 (a) Schematic of the scattering of normally incident plane waves by a cylinder. The right row shows two polarized waves: (1) p–polarization and (2) s–polarization. (b) Real (n) and imaginary (κ) part of the refractive index of c-Si considered in this paper.

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Generally, the angular distributions of scattering patterns can be viewed by the differential scattering cross section dσ^s,p/dθ for s-polarization and p-polarization, respectively. For a cylindrical scatterer, it shows that [52, 53]

As an alternative, we employ the scattering asymmetry parameter g =< cosθ > [14] which is defined as the mean cosine of the scattering angle θ to measure the strength of the scattering directionality. The g related to the photon scattering and transport mean free path is a more general concept in scattering theory [52, 53], which is popular especially for single-particle scattering [54] or light transport in disordered media [55]. The g can be easily calculated based on the differential scattering cross section dσ^s,p/dθ showing [52, 53]

In analogy with analysis of a single sphere studied in [54] taking EDs and MDs into account, the g^s,p can be further specified by inserting Eq. (2) into Eq. (3), showing that

2.2. Realization of strong FS and BS using a thin c-Si cylinder

Let us now investigate the case of a thin cylinder (r = 0.056 µm), where only the first two orders of the Mie expansions contribute to the scattering efficiency under normal incidence. Here we study the scattering properties of a realistic material (c-Si) [33] in the visible regime (0.4 µm ≤ λ ≤ 0.75 µm) whose intrinsic absorption is low with a small imaginary part of optical constants shown in Fig. 1(b).

The results of the scattering efficiency spectra for both polarizations are illustrated in Figs. 2(a) and 2(b) along with the g-value and phase difference Δγ of the first two dipolar modes. Considering effects of EDs and MDs is reasonable here since the contributions of MQs (EQs) or even higher optical modes are relatively small. Clearly, for s-polarized waves, the value of g is always positive showing usual FS features with two representative scattering patterns plotted on the right row. And it is also understandable that the maximum g in points a − d is smaller than $g_{\max }=1/\sqrt{2}$ because the phase difference Δγ is not ideally in-phase. In contrast, in the case of p-polarization, the changes of g are remarkable. The occurrence of the strong negative–g at two points A (λ = 0.41 µm, g^p = −0.29) and C (λ = 0.52 µm, g^p = −0.38), shows fine agreement with the aforementioned BS condition discussed in Sec. 2.1. Besides, we would like to stress that the value of the g studied in the paper is relatively more negative than that of a silicon sphere (λ = 1.55 µm, g = −0.15) in [14]. Though the intrinsic absorption of c-Si in short wavelengths is relatively large, the absorption effects can be totally neglected in the near-infrared regime. Moreover, there is a near-perfect condition with $\left|a_0^p\right|=\sqrt{2}\left|a_1^p\right|$ and Δγ ≈ 0 for strong FS occurring at λ = 0.68 µm, whose BS is well suppressed with g^p = 0.70.

 

Fig. 2 Scattering efficiency spectra of (a) s-polarized waves and (b) p-polarized waves, including total and contribution from EDs, MDs and MQs (EQs). Obviously, the contribution from quadrupole can be ignored reasonably. The right row shows the representative scattering patterns for two polarized waves and the corresponding working wavelengths and the values of scattering asymmetric factor are listed as following: a : λ = 0.43 µm, g^s = 0.43, c : λ = 0.55 µm, g^s = 0.64; C : λ = 0.52 µm, g^p = −0.38, D : λ = 0.68 µm, g^p = 0.7.

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Furthermore, as shown on the right row of Figs. 2(a) and 2(b), as for s-polarization, the scattering pattern at point c shows strong FS, while BS is dominated at point C for p-polarization. Such polarization-dependent directional scattering features also provide a basis to realize some novel polarization-based optical devices, including polarizers or beam splitters. Next, we will show that by utilizing lattice effects based on an proper particle arrangement, a perfect linear polarizer can be achieved when the reflection can be eliminated with zero BS for one polarization and for the other polarization transmission can be suppressed at the same time when the incident waves could interfere destructively with the forward scattering.

3. Design of a perfect linear polarizer through the metalattice

3.1. Theory: Light propagation in 1D metalattices

The 1D infinite periodic metalattice with period d is illustrated in Fig. 3(a) to demonstrate what we propose before. A plane wave with wavevector k (in the x − y plane) incident on the nanocylinders whose radius and optical indices are consistent with those studied in Sec. 2. The incident light under incident angle φ with respect to the x axis can be s-polarization with electric field along the z direction (E₀ ‖ z) and p-polarization with magnetic field parallel to z axis (H₀ ‖ z). The general scattering problem has been analytically solved via multiple scattering theory with the consideration of the lattice couplings. For the j–th nanocylinder centered at R_j on the x − y plane (j = −N : N, 2N + 1 cylinders are considered here), its scattering filed can be expanded into a sum of a series of cylindrical harmonics, and the scattering coefficients ${\tilde{a}}_{jm}^{s,p}$ for the j–th nanocylinder are related to the isolated-nanocylinder scattering coefficients $a_{jm}^{s,p}$ through [52, 56]

 

Fig. 3 (a) Schematic illustration of the wave scattered by the proposed one-dimensional metalattice consisting of c-Si nanocylinders. (b) The extinction ratios for transmission β_T (the left line) and reflection β_R (the right line) in units of decibel (dB) which measure the efficiency of the linear polarization conversion.

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With the expansion coefficients obtained, for the proposed metalattice depicted in Fig. 3(a), the reflection $R_v^{s,p}$ and transmission $T_v^{s,p}$ of the v–th diffraction order can be expressed by [51, 56]

3.2. Design of a perfect linear polarizer

Inspired by the aforementioned design principles, a polarization-dependent metalattice can be achieved in the visible regime (0.4 µm ≤ λ ≤ 0.75 µm) using a row of thin c-Si nanocylinders studied in Sec. 2.2. In this section, we first calculate the extinction ratios for transmission β_T and reflection β_R of two polarized light for 1D arrays with different period d, aiming to find an optimal lattice arrangement where the values of β_T and β_R are extremely high at a certain wavelength. The metalattice contains 2N + 1 = 201(N = 100) nanocylinders, which is sufficient to get reliable results. Here in this paper, we focus on the regime where there is only zeroth-order diffraction in the far field: $k^2-k_{yv}^2<0$ and $R_v^{s,p}$, $T_v^{s,p}=0$ for v ≥ 1. Consequently, the range of the lattice period is restricted within 0 < d ≤ 0.4 µm. Then the transmission $T_v^{s,p}$ and reflection $R_v^{s,p}$ can also be simply expressed as $T_0^{s,p}={\left|1+\frac{2}{d(2N+1)k_x}{\displaystyle {\sum }_{j=-N}^N{\displaystyle {\sum }_{m=-\infty }^{\infty }{\tilde{a}}_{jm}^{s,p}}}\right|}^2$ and $R_0^{s,p}={\left|\frac{2}{d(2N+1)k_x}{\displaystyle {\sum }_{j=-N}^N{\displaystyle {\sum }_{m=-\infty }^{\infty }{(-1)}^m{\tilde{a}}_{jm}^{s,p}}}\right|}^2$. The calculated results of β_T and β_R for different periods and working wavelengths are illustrated together in Fig. 3(b), in which two curves denoting the highest value of β_T (left) and β_R (right) are clearly seen. Therefore, it is possible to choose an optimal lattice period d = 0.32 µm where the β_T and β_R are extremely large in the visible regime (λ ≈ 0.58 µm) simultaneously.

For the metalattice with the optimal period, the transmission $T_0^{s,p}$ and reflection $R_0^{s,p}$ spectra for both polarized waves are shown in Fig. 4(a) along with the results of β_R (top) and β_T (bottom) around λ = 0.58 µm shown in Fig. 4(b). It is clear that transmission can be well suppressed for p-polarization while the reflection is considerably rare for s-polarization, leading to an excellent performance for realization of a polarization-dependent metalattice. Notably, it is reasonable that $T_0^{s,p}+R_0^{s,p}<1$ due to the intrinsic absorption of the c-Si in the visible regime. Still, the extinction ratios for transmission β_T and reflection β_R at λ = 0.58 µm are impressively high with β_T ≈ 17 dB and β_R ≈ 24 dB, which are larger than most of researches studied recently [57, 58].

 

Fig. 4 (a) Numerical results of transmission (red lines) and reflection (blue lines) spectra of p-polarized waves (top) and s-polarized waves (bottom). (b) The extinction ratios for transmission β_T (bottom) and reflection β_R (top), including $T_0^s/T_0^p$ and $R_0^p/R_0^s$ within a certain waveband around λ = 0.58 µm.

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3.3. Analysis of dipolar interferences with lattice couplings

As discussed in Sec. 3.1, due to the lattice couplings, the optical features of the whole metalattice are distinct from the scattering of an isolated nanocylinder. To further clarify the underlying mechanism, firstly, we make a comparison of scattering coefficients between ${\tilde{a}}_{0m}^{s,p}$ of the lattice-nanocylinder and $a_{0m}^{s,p}$ of the individual nanocylinder. According to the Floquet theory [59] ${\tilde{a}}_{jm}^{s,p}={\tilde{a}}_{0m}^{s,p}\exp (i{k}_y{R}_j)$ in a 1D system, because of ${\tilde{a}}_{jm}^{s,p}={\tilde{a}}_{0m}^{s,p}$ with k_y = 0 under normal incidence, it is reasonable to only consider ${\tilde{a}}_{0m}^{s,p}$ here. Similarly, ${\tilde{a}}_{jm}^{s,p}$ also meets ${\tilde{a}}_{jm}^{s,p}={\tilde{a}}_{0m}^{s,p}$ when φ = 0. As illustrated in Fig. 5(a), obviously, for two polarizations at λ = 0.58 µm pointed with p and s in Fig. 4, the lattice couplings play a significant role in improving dipolar excitation efficiency. The magnitudes of the zeroth optical modes (MD for p-polarization and ED for s-polarization) are enhanced over the unity, giving considerable weights to modify scattering patterns of an isolated scatterer. Further, the corresponding far-field angular scattering patterns are illustrated in the right row. The scattering efficiencies at two cases have been dramatically enhanced (the red dot-dashed lines) while the influence of lattice couplings in phases of EDs and MDs is negligible. The directivity of two scattering patterns remain unchanged with only enhancement in scattering amplitude. Actually, it is understandable that, taking s-polarization as an example, the electric field E₀ as well as ED moment is oriented perpendicular to the lattice wave propagation (in the y direction) where the lattice resonances involve [48]. Then, the lattice effects are more prominent for ED reasonably. The analysis can also be applicable for p-polarization to explain why the changes of scattering coefficients for MD is remarkable. On the other hand, by further expanding the differential scattering cross section defined in Eq. (3), one can immediately obtain

 

Fig. 5 The comparison results about the scattering coefficients for an individual meta-atom and the meta-atom embedded in the 1D metalattice with d = 0.32 µm at λ = 0.58 µm, corresponding with the scattering patterns (the right row) for (a) s polarization and (b) p–polarization, respectively. The insets in the left plots of (a) and (b)–illustrate the incident conditions for s-polarization (left, the electric field is perpendicular to the lattice direction) and p-polarization (right, the magnetic field is perpendicular to the lattice direction).

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Furthermore, to make the lattice effects more clear, in Fig. 6(a), we show the transmission and reflection spectra calculated by using the scattering coefficients of an individual cylinder considered in Sec. 2.2. The performance of such metalattices ignoring the effects of lattice couplings shows no desirable polarization-dependent functionality with low transmission for s–polarization and low reflection for p–polarization, further confirming the role of the coupling effects which ensures the large improvement in polarization conversion efficiency discussed in Sec. 3.2. Besides, the lattice couplings also have some influences in modifying scattering angular distributions. The comparison results of g^s,p for both the single cylinder and the cylinder in the lattice are illustrated in Fig. 6(b). Though the changes of g^s,p are not large consistent with the analysis mentioned above, the reduction of g^p at short wavelengths is also remarkable, which is crucial in increasing the reflection for metasurfaces due to the existence of lattice effects. The representative scattering pattern pointed with e, as a inset in Fig. 6(b), shows that the BS has been further enhanced at λ = 0.5 µm where the value of g^p is decreased from −0.35 to −0.52, which can potentially improve the polarization conversion efficiency.

 

Fig. 6 (a) The transmission and reflection spectra for both polarized waves calculated with the scattering coefficients of an isolated cylinder. (b) The comparison results of the scattering asymmetry factor g^s,p based on the scattering coefficients of an isolated cylinder (the full blue lines) and the lattice cylinder (the dashed red lines). The inset in the top plot in (b) represent the scattering patterns at the spectral position denoted with e.

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In fact, even though the only zeroth diffraction is included in the far field, lattice couplings with dipolar interferences can still be affected due to the existence of high-order diffractive evanescent field. In the near field, as for the first diffraction order (v = 1), $k_{x1}^2=k^2-k_{y1}^2<0$ and the lattice wave vector becomes imaginary. It is possible that the evanescent filed of high-order diffractions can interplay with dipolar modes within a single meta-atom under an appropriate lattice period, which ensures the high-efficient couplings. In Fig. 7, we illustrate the electric field intensity |E|² with three lattice periods at λ = 0.58 µm to show the difference of the lattice effects in resonance features. When the period is small with d = 0.2 µm shown in Figs. 7(a)(1) and 7(b)(1), although the lattice effects may appear, the field enhancement is relatively weak or mainly occurs in free space (air) [60], showing weak effects in enhancing scatterer resonances. Similarly, in the case of large period (d = 0.4 µm), the role of lattice effects is decreased and the scattering profile is more akin to the single scattering shown in Fig. 7(a)(3). While for the optimal lattice arrangement (d = 0.32 µm) calculated in the last part, a large field enhancement can be observed in the vicinity of the cylinders with the brightest color map illustrated in the middle row of Fig. 7. These changes will be further verified in Sec. 4.1. Actually, there is a trade-off between the decay of high-order diffraction modes and lattice-coupling strength in our systems, thus leading to an proper arrangement with desirable performance. Consequently, we can conclude that the scattering coefficients can be significantly enhanced due to the near-field couplings between dipolar interferences and lattice-induced high-order diffractive evanescent field under a proper lattice period.

 

Fig. 7 Electric field intensity for (a) p–polarization and (b) s–polarization at three different lattice periods, including (1) d = 0.2 µm, (2) d = 0.32 µm and (3) d = 0.4 µm at λ = 0.58 µm. Three subplots share the same color bar. The red single arrows show the incident direction and the white double arrows denote different lattice periods.

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Therefore, based on the design principles and detailed analysis in this section, the lattice effects on dipolar interferences have been distinguished unequivocally comparing with the single-particle scattering. The lattice couplings play a vital role not only in improving the efficiency of the metalattice but also in directional modification of scattering patterns within certain wavelengths, which suggests a new way to design and further manipulate functionalities of the metasurfaces. In the next section, we would like to show that the proposed polarizer exhibits robust and excellent performance when considering practical implementation issues.

4. Discussion on practical implementation

Above analysis indicates that the lattice effects in dipolar interferences are of great importance and the calculated results about perfect transmission, reflection and large extinction ratios of the metalattice further confirm the reliability of our design principles proposed in this paper. In this section, we further discuss some inevitable problems in practical implementation of the designed metalattice, especially for the tolerance of lattice imperfections, the effects of the system size, the dependence on incident angles and the applicability for different working wavelengths as well.

4.1. Tolerance of lattice imperfections

In practical fabrication, it is inevitable to suffer some lattice imperfections including the nonuniformity in diameters of the nanocylinders, the irregular shapes of the cross sections, small but non-negligible disorder in metaatom positions and so on [58]. Here in this part we mainly focus on the disordered perturbations in cylinder arrangements and further indicate the central role of perfect lattice effects. The perturbation Δd is introduced to measure the position deviations from the perfect lattice arrangements. The range of the perturbation Δd is chosen to be $\left|\mathrm{\Delta }d\right|\le \frac{1}{2}(d-2r)$, aiming to avoid the overlap between adjacent nanocylinders. Then the disordered degree can be defined as χ = |Δd|/|Δd|_max ranging with 0 ≤ χ ≤ 1.

In Fig. 8, we present how the functionalities of the proposed polarizers (λ = 0.58 µm, period d′ = d + Δd) are influenced by lattice disorder, in which all the results are calculated by assemble averaging 50 different configurations. It can be seen in Fig. 8(b) that both β_T and β_R experience apparent decreases as the disordered degree increases. Even though the performance of the polarizer with lattice imperfections are damaged compared with the perfect periodic counterpart, the functionality can still be preserved with β_T > 10 dB and β_R > 20 dB for χ ≤ 0.6. When the disordered degree increases to χ = 1, the extinction ratios drop to β_T ≈ 8 dB and β_R ≈ 17 dB which are still acceptable in practical implement. Notably, in Fig. 8(a), we also show the absorption for two polarizations with $A_0^{s,p}=1-T_0^{s,p}-R_0^{s,p}$. The disorder imperfections introduced here play important roles in reducing transmission $T_0^s$ for s–polarization and weakening reflection $R_0^p$ for p–polarization but few significant change in $T_0^p$ and $R_0^s$, leading to the considerable absorption enhancement as the disordered degree increases [61]. Besides, the absorption enhancement for p polarization is larger than that of s polarization, since for p-polarized waves electric fields are in the x − y plane and multiple scattering can be further strengthened ascribed to the occurrence of disorder in lattice.

 

Fig. 8 Influence of the disordered imperfections on the performance parameters, including (a) transmission $T_0^{s,p}$, reflection $R_0^{s,p}$ and absorption $A_0^{s,p}=1-T_0^{s,p}-R_0^{s,p}$; (b) extinction ratios for transmission β_T and reflection β_R. χ is defined as the disordered degree. χ = |Δd|/|Δd|_max. (c) Influence of the disordered imperfections in the coupling strength η^s,p as a function of the lattice periods at λ = 0.58 µm.

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Meanwhile, we further investigate how the strength of lattice couplings responses to the lattice imperfections. According to the analysis in Sec. 3.3, results indicate that lattice effects play a dominating role in improving scattering efficiency of ED for s-polarization and MD for p-polarization. Here we define a coupling strength η^s,p to measure the strength of lattice couplings with dipoles showing

4.2. Effect of system size

Above numerical calculations are performed with a metalattice consisting of 2N + 1 = 201 (N = 100) cylinders. Since the existence of edge effects in finite systems, it is necessary and as well as of practical significance to further investigate the dependence on system size, evaluating whether a larger or smaller metalattice can still perform well with the same lattice period.

In Fig. 9, we show how the performance parameters of the proposed linear polarizers vary with the sizes of the metalattices, where the calculation is conducted to the largest system with 201 nanocylinders. Obviously, the transmission $T_0^{s,p}$, reflection $R_0^{s,p}$ and the performance of extinction ratios for transmission β_T and reflection β_R are all convergent in the end. It can be observed that for the systems containing over 41 (N = 20) nanocylinders, the performance (β_T and β_R) of the metalattices can be well preserved, indicating that the designed linear polarizer can still operate for large-size systems. While for small cases, the influences of size effects in β_T and β_R are significant with a large fluctuation especially for extremely small systems. In the case of the N = 10 shown in Fig. 9, the extinction ratios for transmission β_T ≈ 17 dB remains relatively stable, while the β_R witnesses a sharp increase with 40 dB where the edges effects are more substantial and the Floquet theory may be not valid. Results suggest that careful modification of the lattice period should be considered when metalattices operating with small sizes. Even so, as for all the cases calculated here, the extinction ratios for transmission β_T and reflection β_R can still be ideally higher with β_R > 18 dB and β_T > 15 dB, ensuring the excellent and reliable performance in practical implement.

 

Fig. 9 The performance parameters, including transmission $T_0^{s,p}$, reflection $R_0^{s,p}$, extinction ratios for transmission β_T and reflection β_R change with the system sizes. The perfect functionality can be well preserved as long as the value of N is not below 20.

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4.3. Acceptance of incidence angles

In this part, we study the effects of oblique incidence on performance of the designed metalattices possessing excellent polarization dependent features. Since we merely focus on the diffractionless regime, the range of incident angles of interest is restricted within $\phi \le \arcsin \left(\frac{d}{\lambda }-1\right)=54°(d=0.32\mu \mathrm{m},\lambda =0.58\mu \mathrm{m})$ where only zeroth order diffraction should be considered. When φ is larger than 54°, higher multipoles may be excited and those can interference with each other as well as with lattice couplings in the far field, leading to the generation of higher diffraction orders, which is not taken into account in this paper.

The results of the extinction ratios for transmission β_T and reflection β_R at λ = 0.58 µm varying with the incident angles are illustrated in Fig. 10(b) along with the reflection and transmission spectra depicted in Fig. 10(a). It is clear that under oblique incidence, the performance of the metalattice is unavoidably destroyed to some extent. When the incident angle is smaller than around 10°, the optical response of the polarizer is reliable with both β_T and β_R larger than 10 dB. The acceptance of the incident angles for the 1D metalattice can be attributed to the high refractive index contrast between nanocylinders (c-Si) and background (air). On the other hand, the nanocylinders are subwavelength in diameter, which also reduces the shadow effects from adjacent metaatoms [62]. But the significant decrease for large oblique angles cannot be ignored in Fig. 10(b), suggesting that the incident conditions should be well controlled within the acceptance range, or lattice period and radius of nanocylinders should be further modified to well preserve the performance of the designed polarizer.

 

Fig. 10 The performance parameters, including (a) transmission $T_0^{s,p}$ and reflection $R_0^{s,p}$; (b) extinction ratios for transmission β_T and reflection β_R change with the incident angles.

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4.4. Realization of polarizers with different working wavelengths

According to the design principles discussed in this paper, reliable functionalities of the linear polarizers can be achieved. Since the perfect directional conditions are considered here, the bandwidth of the metalattice-based polarizer is not broad seen from Fig. 4(b). Nevertheless, the working wavelengths can be tuned by properly changing the lattice period and the geometric parameters of nanocylinders. Next, we will show for different target working wavelengths the linear polarizers can be also achieved in the visible range. The obtained desirable extinction ratios for transmission β_T and reflection β_R as well as corresponding the optimal lattice period d and cylinder’s radius r are illustrated in Fig. 11.

 

Fig. 11 (a) The optimal values of the lattice period d and the radius of cylinders r for different working wavelengths, along with (b) the corresponding extinction ratios for transmission β_T and reflection β_R at each working wavelength.

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In terms of different target wavelengths, as shown in Fig. 11(a), it is possible to find appropriate parameters with an optimal period d and a radius r of cylinders based on the numerical analysis in Sec. 3.2. The extinction ratios for λ > 0.55 µm considered here are considerably large with β_T, β_R > 15 dB, which are desirable in practical applications. Besides, it can be seen that as the working wavelength increases, the optimal lattice period and the radius of cylinders also increase linearly, which agrees well with the scale invariance rule of Maxwell equations since the optical constants within 0.55 µm < λ < 0.75 µm are almost similar shown in Fig. 1(b). Note that the results when λ < 0.55 µm are not illustrated here because the intrinsic absorption of c-Si in short wavelengths is too large, thus the polarization characteristics are destroyed inevitably. In contrast, the absorption loss among long wavelengths is negligible and the outstanding performance can be preserved, which still offers a promising way to design precise optical devices.

5. Conclusion

In this work, we propose the design principles to achieve excellent directional polarization manipulations of an individual meta-atom and metelattices by taking the dipolar interferences and lattice couplings into consideration. Firstly, we employ the scattering asymmetry factor g to investigate the scattering directionality of an individual cylindrical particle in the visible regime and find the extremum value of the $g(|g|)\le 1/\sqrt{2}$ for a cylinder. The strong forward (g^s = 0.64; g^p = 0.70) and backward (g^p = (0.38) scattering can be observed in two different polarized waves simultaneously owing to the efficient excitation of EDs and MDs.

By combining lattice effects and dipolar interferences, we propose a metalattice-based linear polarizer at the visible wavelength stimulated by the aforementioned polarization-dependent phenomena. After properly designing the lattice period d = 0.32 µm, the transmission light through the designed metalattice can be perfectly vanished for p-polarization and for s-polarization reflection is also eliminated, showing excellent polarization-selective performance with the large extinction ratios (β_T ≈ 17 dB, β_R ≈ 24 dB). Furthermore, we have theoretically discussed the underlying mechanism for its feasible functionality. Analyses show that the efficiency of dipolar excitations is enhanced significantly ascribed to the interplays between the dipolar modes and high-order diffractive evanescent field with an appropriate lattice arrangement, leading to a considerable modifications of scattering patterns of an isolated scatterer. Besides, we also reveal that for 1D cylindrical systems, the connections between dipolar interferences and lattice couplings show obvious polarization selectivity. As for s-polarized waves, the lattice couplings mainly affect the profile of ED, while such influence in MD is prominent for p-polarization. On the other hand, the proposed linear polarizer shows stable and reliable performance in practical implement. The large extinction ratios for transmission β_T and reflection β_R can be preserved despite of the existence of lattice imperfections, while the absorption is enhanced as the disordered degree increases, which also indicates that the optimal lattice couplings emerge in the case of perfect lattice arrangement (χ = 0). In addition, the insensitivity to the system sizes (2N + 1 > 41) and the acceptance of the incident angles (φ < 10°) of our metalattice are also attractive and desirable in many applications. Furthermore, we also show that polarizers with different working wavelengths can be also obtained by tuning the lattice period and the radius of cylinders. The theory and designing principles in this paper provide a promising platform to investigate the effects of dipolar interferences and lattice couplings, which can also potentially stimulate more novel optical phenomena and applications.