Compact and planar optical beam splitters are highly desirable in various optical and photonic applications. Here, we investigate two kinds of optical beam splitters by using oligomer-based metasurfaces, one is trimer-based metasurface for 3-dB beam splitting, and the other is pentamer-based metasurface for 1:4 beam splitting. Through electromagnetic multipole decomposition and in-depth mechanism analyses, we reveal that the electromagnetic multipolar interactions and the strong near-field coupling between neighboring nanoparticles play critical roles in beam-splitting performance. Our work offers a deeper understanding of electromagnetic coupling effect in oligomer-based metasurfaces, and provides an alternative approach to planar beam splitters.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical beam splitters, which split a beam of light into two or more predesigned directions, have wide applications in interferometers, multiplexers, dual/multi-channel sensing and optical nanocircuitry. High-performance beam splitters are well implemented utilizing various methods and structures, such as diffraction gratings [1–3], photonic crystals , multimode interference couplers , directional couplers  and phase gradient metasurfaces [7–9]. In this paper, we investigate an alternative approach to planar optical beam splitters by using oligomer-based metasurfaces.
Oligomer-based metasurfaces have attracted a growing attention in recent years due to the rich electromagnetic multipolar interactions and strong near-field coupling effect inside their structural units [10,11]. Their structural units are oligomers, which are composed of two or more subwavelength metallic or dielectric particles (or optical scatters) with subwavelength separations between them, therefore, strong near-field coupling effect inherently exists inside an oligomer . According to the number of constituent particles in an oligomer, the structural unit of oligomer-based metasurfaces can be a dimer, a trimer, a tetramer, a pentamer, and so on.
Dielectric oligomer-based metasurfaces, whose structural unit consists of a few dielectric particles, have much lower optical losses than their metallic (or plasmonic) counterparts, especially at optical frequencies. According to Mie scattering theory , a dielectric nanoparticle with approximately designed dimensions not only supports electric dipole (ED) and magnetic dipole (MD), but also excites higher-order dipoles, such as electric quadrupole (EQ) and magnetic quadrupole (MQ), and even toroidal dipole (TD) [13–16]. These electric and magnetic resonance modes and their interplay, along with the inherently strong near-field coupling effect inside the oligomers, enable all-dielectric oligomer-based metasurfaces more degrees of freedom to realize interesting optical responses or unique optical functionalities. For example, oligomer-based metasurfaces have been demonstrated for Fano resonance , electromagnetically induced transparency  and optical anapole mode ; and they have also been applied in beam deflection , nonlinear effect enhancement , electric and magnetic hotspots .
Different from these reports, we proposed a trimer-based metasurface for 3-dB beam splitting and a pentamer-based metasurface for 1:4 beam splitting , and in this paper, we further make an in-depth investigation to reveal the behind operation mechanism by using electromagnetic multipolar decomposition method. We find that the interactions of different electric and magnetic resonance modes and strong near-field coupling play critical roles in beam-splitting. In addition, we analyze the beam splitting performance deeply and discuss the effects of symmetric and asymmetric near-field coupling.
In comparison with commercially available diffractive beam splitters, our designed devices have disadvantages in terms of performance, design and fabrication. However, our work presents an alternative approach to planar optical beam splitters, and offers a deeper understanding of electromagnetic coupling effect in oligomer-based metasurfaces.
The rest of the paper is organized as follows: section 2 introduces the device structure, working mechanism and beam-splitting performance for the 3-dB beam splitter, whereas section 3 introduces those for the 1:4 beam splitter. Brief conclusions are given in the final section.
2. Device structure and operation principle
2.1 Device structure
The basic structural unit of the 3-dB beam splitter is a trimer nanoantenna as shown in Fig. 1, which consists of a trimer on top of a silica (SiO2) substrate. The trimer is composed of three silicon (Si) nanocylinders, that is, one bigger subwavelength Si cylinders (radius R1=250 nm) and two smaller Si cylinders (radius R2=200 nm). All the three Si nanocylinders have the same height h=310 nm, and the separation between them is d=10 nm. Multiple trimer structural units are periodically arranged to form a trimer array, i.e., a trimer-based metasurface. The periods along the x- and y-directions are Lx=1950 nm and Ly=550 nm, respectively. Unless otherwise specified, these parameter values will not change throughout this paper. An x-polarized wave normally illuminates onto the metasurface along the positive z-axis.
In order to explain the working mechanism of the 3-dB beam splitter, we begin with analyzing the directional scattering of a single trimer, because it is the structural unit of the trimer-based metasurface, and its scattering behavior is the basis of beam-splitting performance.
2.2 Directional scattering property of a single trimer
As we know, an individual dielectric nanoparticle, such as a nanosphere, a nanocylinder or a nanodisk, may simultaneously support electric and magnetic resonance modes, and the interference effect between their radiation fields can be adjusted by changing the size, shape of nanoparticles, and then the scattering ratio between forward and backward directions can be controlled . Two nanoparticles with different sizes form an asymmetric dimer, and their near-field coupling effect can be utilized to realize tunable directional scattering or light routing by changing the wavelength of incident light [20,24,25].
To clarify the beam-splitting mechanism of the trimer-based 3-dB beam splitter, we begin with analyzing the directional scattering of a single Si nanocylinder (R=250 nm) and that of an asymmetric dimer consisting of two nanocylinders with different radii (R1=250 nm, R2=200 nm), whose cross-sectional views are shown in Figs. 2(a1) and 2(a2), respectively. The corresponding far-field scattering patterns at wavelength λ=1141 nm are obtained by using finite-element method (COMSOL Multiphysics software) and shown in Figs. 2(b1) and 2(b2), respectively. As can be seen, the single Si nanocylinder and the asymmetric dimer display different scattering behaviors, the former realizes good forward scattering, whereas the latter exhibits oblique scattering behavior, deflecting the light to the right side.
From above, we guess that, if two smaller Si nanocylinders are symmetrically arranged on both sides of the larger one to form a trimer as shown in Fig. 2(a3), then the light beam can be symmetrically scattered to the left and right sides. This presumption is verified by its far-field scattering pattern in Fig. 2(b3), the transmitted beam is indeed deflected to the left and right sides with the same angle.
Now, we investigate in depth why the trimer possesses such directional scattering property. According to Mie theory, when the size of a dielectric nanoparticle is comparable to the wavelength of incident light, the total scattering field can be decomposed into the scattering fields of electric and magnetic modes with different orders . Here we only consider electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ) and magnetic quadrupole (MQ), because the sizes of the nanocylinders are much smaller than operation wavelength (1000 nm-1600 nm), higher-order electric and magnetic resonance modes can be ignored due to the extremely weak influence on the total scattering.
According to Eqs. (5)–(8), we calculate the scattering spectra of different electric and magnetic modes by using finite element method and electromagnetic multipole decomposition method, and show the results in Fig. 3. It can be seen that ED and MD modes have dominant contributions to the total scattering; whereas the EQ mode has a resonance peak at 1050 nm, but its scattering intensity is quite low at other wavelengths; the scattering intensity of MQ mode is negligible compared with other modes in the whole waveband. Therefore, we only need to consider ED and MD modes and their interaction.
When an x-polarized plane wave normally illuminates onto the trimer antenna along the positive z-axis, each of the three nanocylinders will be excited to produce an ED mode along the x-direction and a MD mode along the y direction [29–31]. For convenience, we mark them as ED1x, ED2x and ED3x, MD1y, MD2y and MD3y, in which the subscript 1, 2 and 3 stand for the three nanocylinders, as shown in Fig. 4(a). These electric and magnetic dipoles are induced by the incoming light. Apart from these, there are other modes originating from the interaction between neighboring nanocylinders. In the trimer, the three nanocylinders are very close to each other, with deep-subwavelength separation between them. Therefore, the middle larger nanocylinder strongly couples with the two smaller ones, generating an ED component parallel to the light incident direction , namely, ED1z, ED2z and ED3z, as shown in Fig. 4(a). Later on, we will clarify that these coupling-effect-induced dipoles play a critical role not only in the directional scattering property of a single trimer, but also in the trimer-based metasurface.
Figure 4(b) plots the amplitudes of ED1z, ED2z and ED3z versus wavelength, and the amplitude of ED1x is also given for comparison. As can be seen, ED2z is nearly zero, indicating almost no EDz component is generated inside the middle larger nanocylinder. However, the amplitudes of ED1z and ED3z are always equal to each other, and they have the same order of magnitude as ED1x, indicating that effective and equal EDz modes are generated inside the two smaller nanocylinders. Figure 4(c) shows the real parts of ED1z and ED3z as a function of wavelength. As can be seen, they are always with opposite signs, meaning that ED1z and ED3z always have opposite directions, or they are π radians out-of-phase. In contrast, ED1x, ED2x and ED3x have the same directions (not shown here), therefore, for the convenience of simplification, we use EDx to represent them. And because of the similar reason, we use MDy to stand for MD1y, MD2y and MD3y.
In order to gain a deep understanding of the interactions among EDx, EDz and MDy and their influence on the directionality of scattered light, we present 2D far-field scattering patterns of these modes in the first row in Fig. 5, according to the method in reference . And the total scattering field of the trimer will be determined by the overlap of scattering fields of these modes. It needs to be stressed that EDx and EDz modes not only interact with the MDy mode of the same nanocylinder, but also interact with the MDy mode of the neighboring nanocylinders.
Firstly, we consider the interaction of EDx and MDy, which can be represented by process I in Fig. 5. When the phase difference between them is in the region of (-π/2, π/2), the two modes are considered to be in-phase, and the forward scattering intensity (along the positive z- direction) is much higher than the backward scattering intensity (along the negative z-direction), as can be seen from the middle subgraph in the second row of Fig. 5.
Apart from the above mentioned forward scattering, side scattering is also required to turn the scattered light to the left or right side [28,29,32,33]. Fortunately, the interaction of EDz and MDy can realize side scattering. When their phase difference is in the region of (-π/2, π/2), side scattering mainly occurs in the negative x-direction; and when it is in the region of (-π, -π/2) or (π/2, π), side scattering mainly occurs in the positive x-direction. Thus, when ED1z and ED3z interact with MDy (see processes II and III in Fig. 5), one side scattering is turned to the negative x-direction, the other is to the positive x-direction, as can be seen from the left and right subgraphs in the second row of Fig. 5. This is because ED1z and ED3z are π radians out-of-phase, as mentioned above.
Then, the combination of forward scattering and negative x-direction side scattering will make the scattered light obliquely turned to the left (see process IV in Fig. 5); while the combination of forward scattering and positive x-direction side scattering will make the light obliquely turned to the right (see process V in Fig. 5). Finally, the resultant far-field scattering displays symmetric scattering pattern, the light is evenly deflected to the left and right sides, as indicated in the third row of Fig. 5.
In the above, the two smaller nanocylinders are symmetrically set on the two sides of the larger one (i.e., d1=d2=100 nm), and they have equal coupling strength with the larger nanocylinder, i.e., ED1z=ED3z. Therefore, the far-field scattered light is symmetrically deflected. If the two smaller nanocylinders are distributed asymmetrically, how does the unequal coupling strength affect the far-field scattering?
In the following, by setting d2=100 nm and d1=10 nm, we investigate the effect of asymmetric distribution on far-field scattering. The cross-sectional view of this trimer is shown in Fig. 6(a), and the amplitudes of ED1z and ED3z versus wavelength are presented in Fig. 6(b). In most wavelength ranges, ED1z is much stronger than ED3z, i.e., ED1z>>ED3z, meaning that the middle nanocylinder couples to the left one with greater strength than it to the right one.
We further obtain the far-field scattering pattern of the trimer at λ=1141 nm and show it in Fig. 6(c). Obviously, the trimer exhibits asymmetric scattering behavior, the transmitted light is obliquely deflected to the left, without light scattered to the right. The above results mean that asymmetric coupling strength will produce an asymmetric directional scattering. Based on this, we can design beam splitters with different split-ratios by changing d1 or d2. Later on, we will discuss this in detail.
Now let’s make a brief summary to this subsection. By using Mie theory and electromagnetic multipole decomposition method, we reveal that the near-field coupling effect between neighboring nanoparticles plays a critical role in the directional scattering property of a single trimer, because the coupling-effect-induced dipoles offer the indispensable side scattering for the light to be deflected into two or more directions. Besides, we find that asymmetric spatial arrangement of the constituent nanoparticles will result in asymmetric coupling strength, and hence an asymmetric directional scattering.
In the next subsection, we will further investigate the effect of near-field coupling on the beam-splitting performance of the trimer-based metasurface.
2.3 Performance analyses of the 3-dB beam splitter
2.3.1 Beam-splitting performance
In the above, we have made a detailed qualitative analysis on the directional scattering behavior of a single trimer. Now, we periodically arrange them to form a trimer-based metasurface, and perform full three-dimensional finite-difference time-domain (FDTD) simulation by employing the FDTD solutions from Lumerical, in which period boundary conditions along the x- and y-directions and perfectly matched layer condition along the z-direction are applied.
The far-field property of the trimer-based metasurface is different from that of the single trimer. As indicated previously (referring to the third row of Fig. 5), when the single trimer is under normal illumination of x-polarization waves, the transmitted light is simultaneously deflected to the left and right sides with the same intensity and at the same angle. This is to say, the maximal transmitted intensity is located at ±1st diffraction orders, with very weak intensity at the 0th order, as shown in Fig. 7(a).
When multiple trimers are periodically arranged to form the trimer-based metasurface, things become different because of interference effect. The interaction of multiple trimers can be equivalent to multi-beam interference, as shown in Fig. 7(b).
The final result will be the combined effect of multi-beam interference and the directional scattering behavior of the single trimer. Importantly, due to the modulation of multi-beam interference, the angular width of the diffracted beam becomes much smaller, in sharp comparison with that of the single trimer. Therefore, the trimer-based metasurface exhibits better directionality, as shown in Fig. 7(c).
In other words, the trimer array (i.e., the trimer-based metasurface) can be viewed as a metagrating, a special grating. The diffraction angles of different orders will follow grating equation, but the light intensities at different orders are heavily dependent on the directional scattering property of the single trimer, which is governed by the near-field coupling effect between the constituent nanoparticles in the trimer.
By performing FDTD simulation to the trimer-based metasurface, we obtain and plot deflection angles and light intensities as a function of wavelength in Fig. 8(a) and transmittance spectra in Fig. 8(b). We find that the deflection angles of the two split beams, i.e., m= -1 and m= +1 orders, indeed obey grating equation:
Besides, in the wavelength ranges of 1141 nm∼1238 nm nm and 1309 nm∼1436 nm (marked by the vertical dotted lines in Fig. 8(a) and the pink areas in Fig. 8(b)), most of the transmitted light energy is symmetrically concentrated on m= -1 and m= +1 orders. Especially within 1309 nm∼1436 nm, the transmittances at m= -1 and m= +1 orders are both above 35%, while the transmittance at m=0 order is below 10%. This indicates that 3-dB beam splitting efficiency is obtained at a wide wavelength range.
We also simulate the transmittance spectra when y-polarized waves normally illuminate onto the metasurface (See Fig. 13 in Appendix 1). The simulation result shows that m= ±1 orders have much lower transmittances than m = 0 order in most wavelength ranges, meaning that the designed device no longer exhibits 3-dB beam-splitting ability for y-polarization waves. Therefore, it is a polarization-dependent beam-splitter.
At λ=1141 nm, Fig. 9(a1) shows that the electric field intensities inside the three nanocylinders are quite low, in sharp contrast to the significant electric-field enhancement in the gap between neighboring nanocylinders, which means that strong near-field coupling effect occurs between them. According to the previous analysis, EDz modes will be generated inside the two smaller nanocylinders with equal strengths, resulting in symmetric side scattering. Therefore, transmitted beams at m= ±1 orders have much higher light intensity than m=0 order, and the trimer-based metasurface shows 1:2 (i.e., 3 dB) beam-splitting effect, as shown in Fig. 9(b1). At λ=1398 nm, the electric field and the transmitted intensity distributions are similar to the situation at λ=1141 nm, as shown in Figs. 9(a3) and 9(b3).
Now we observe the situation at λ=1284 nm. Figure 9(a2) shows that almost no electric field distributes in the gap between neighboring nanocylinders, implying a very weak coupling effect; but there is a strong electric dipole resonance inside the middle nanocylinder, it will consume the majority of light energy, thereby the light intensities at all diffraction orders are very low, as shown in Fig. 9(b2).
To briefly conclude, at some wavelengths, strong near-field coupling effect takes place between neighboring nanoparticles, this endows the trimer-based metasurface the ability of beam splitting. Whereas at other wavelengths, near-field coupling effect is very weak, and electromagnetic wave energies are mostly consumed by electric or magnetic resonance inside nanoparticles, and the light intensities at all diffraction orders are very low, hence the metasurface is not suitable for beam splitting at these wavelengths.
In the above, the middle nanocylinder has equal separations from the left and right ones, that is, d1=d2=d=10 nm, and the trimer-based metasurface exhibits 3-dB beam splitting in a broadband spectral range. Because near-field coupling effect strongly depends on the separation between the nanoparticles, one can’t help wondering how the metasurface will perform when the separation is changed.
2.3.2 Effect of near-field coupling: symmetric coupling and asymmetric coupling
Firstly, we discuss the symmetric coupling case when d1=d2. In this case, the two smaller nanocylinders are symmetrically distributed on both sides of the larger nanocylinder, which will lead to equal coupling strengths and thus the m= -1 and m= +1 orders will have equal transmittances, i.e., T-1 = T+1.
This can be verified in Fig. 10, which plots the transmittances of different orders at λ=1141 nm as a function of separation d. It shows T-1 always equals T+1. With the increase of d, both of them and the total transmittance are reduced, while the transmittance of m=0 order, T0, is increased. This is because coupling effect becomes smaller, EDz modes generated inside the two smaller nanocylinders become too weak to realize effective side scattering, hence m=0 order diffraction (i.e., forward scattering) will predominate over the other two orders for larger d, and then the beam splitting performance will become worse. This also proves that the near-field coupling effect between neighboring constituent nanoparticles plays a critical role in beam splitting.
Now, we discuss the asymmetric coupling case when d1≠d2. In this case, the two smaller nanocylinders are asymmetrically distributed on both sides of the larger nanocylinder, resulting in asymmetric coupling strength and thus asymmetric beam splitting effect, with T-1≠T+1. Here, we define split ratio (SR) as:
By varying d2 while keeping d1=10 nm unchanged, we investigate the effect of asymmetric distribution on split ratio, and present the result in Fig. 11, which plots the transmittance of different orders as a function of d2 at λ=1141 nm. As can be seen, with the increase of d2, T0 is always below 4%, and T-1 is firstly increased up to 42% and then nearly maintains stable, while T+1 is gradually reduced. Consequently, the split ratio is gradually increased from 1:1, and finally reaches 6:1. However, the deflection angles of the two split beams keep unchanged at 35.3° (not shown here). This is because the diffraction period Lx keeps constant, according to diffraction Eq. (9), the deflection angles will keep unchanged.
On the other hand, Fig. 11 also shows the bad tolerances of the designed devices to fabrication errors. In manufacturing process, it is difficult to make the spacings completely consistent with the designed values, any small asymmetry in the gaps will result in a significant imbalance of the beam splitting ratio. In Appendix 2, we make a detailed discussion to show how different variations in different cells impact the performance of beam-splitters.
3. Device structure, operation principle and performance of 1:4 beam splitter
Here, we combine one Si nanocylinder of radius R1=250 nm with four Si nanocylinders of radius R2=200 nm to form a pentamer, as shown in Fig. 12(a). All of the five nanocylinders have the same height h=310 nm. In order to induce strong coupling effect between neighboring nanocylinders, the separation between the middle larger nanocylinder and the smaller ones is still set as d=10 nm. When multiple pentamers are periodically arranged, one gets a pentamer-based metasurface.
According to the working mechanism of the trimer-based metasurface, we anticipate that the pentamer-based metasurface can deflect a normal incident beam into four beams in the transmission space. As shown in Fig. 12(b), when a 45° linearly polarized wave normally illuminates the metasurface along the positive z-axis, four beams of transmitted light appear in the far-field, and they are symmetrically and evenly distributed in four directions with equal transmission intensities.
We define the specular transmitted light as (0, 0) order (the diffraction angle is 0°). The two orders distributed along the x-direction are named (1, 0) and (-1, 0) orders, and their deflection angles are found to be determined by horizontal period Lx (Lx=1480 nm), satisfying grating equation Lxsinθ=mλ (m= -1/+1). The two orders distributed along the y-direction are named (0, 1) and (0, -1) orders, and their deflection angles are found to be determined by vertical period Ly (Ly = 1480 nm), meeting grating equation Lysinβ =nλ (n= -1/+1).
Figure 12(c) maps the transmittance spectra for all transmission orders and the total transmittance. As can be seen, in the wavelength range 1041 nm∼1073 nm (as marked by the pink area in Fig. 12(c)), the total transmittance exceeds 65%, while the transmittance for (0,0) order is less than 5%, and the other four orders have equal transmittance values, all reaching 16%. Particularly, the transmittance at λ=1080 nm for (0, 0) order is as low as 1%, while the transmittances for the other four orders all reaches 17%. Therefore, the pentamer-based metasurface can concentrate light energy into four desired transmission orders, performing 1:4 beam splitting function.
However, it is worth stressing that the designed 1:4 beam splitter and the 3-dB beam splitter are very difficult to fabricate under their currently chosen parameters because the 10 nm spacing between the Si nanocylinders is too small to resolve with standard tools, including e-beam lithography. In addition, the aspect ratio (310 nm: 10 nm = 31:1) is also challenging for the silicon etching process. Therefore, further optimization in structure and parameters are highly desired.
Oligomer-based metasurfaces have more degrees of freedom to manipulate electromagnetic waves owing to the strong near-field coupling effect between its constituent neighboring nanoparticles. In this paper, we investigate two kinds of optical beam splitters by using all-dielectric oligomer-based metasurfaces, one is trimer-based metasurface for 3-dB beam splitting, and the other is pentamer-based metasurface for 1:4 beam splitting. Simulation results shows that both of them are able to realize broadband beam-splitting at near-infrared frequencies. Through electromagnetic multipole decomposition and in-depth mechanism analyses, we find that coupling-effect-induced dipoles enable the indispensable side scattering for the light to be deflected into two or more desired directions, endows the oligomer-based metasurface the ability of beam splitting. Moreover, we find that asymmetric spatial arrangement of nanoparticles in the oligomer leads to asymmetric coupling and thus unequal intensities for the split beams, and the intensity ratio can be readily controlled by the separation of the nanoparticles.
Our proposed beam splitters differentiate from those using phase gradient metasurfaces in that they do not rely on phase gradient, thereby overcoming the limitation of phase-mapping approach. More importantly, it is also fundamentally different from traditional grating-based beam splitters. The oligomer-based metasurfaces can be viewed as metagratings, the transmission angles of the split beams still follow grating equation, however, the light intensities at different orders are heavily dependent on the near-field coupling effect between the neighboring constituent nanoparticles in the unit cell. In comparison with commercially available diffractive beam splitters, our designed beam splitters are polarization sensitive and have lower efficiency. Moreover, under the currently chosen structure and parameters, the proposed devices are difficult to fabricate due to the small spacings, high aspect ratio and small tolerance to fabrication errors. Therefore, further efforts should be made to optimize device structure and parameters. However, despite these limitations, we believe our work provides a deeper understanding of electromagnetic coupling effect in oligomer-based metasurfaces, and offers an alternative approach to planar beam splitters. Besides, it could inspire further development for other novel oligomer-based metasurfaces based on similar principles.
Figure 13 shows what happens when a y-polarized wave normally illuminates onto the trimer-based metasurface.
This appendix discusses the impact of different variations in different cells on the beam-splitting performance of the trimer-based metasurface.
It is difficult to make the spacings completely consistent with the designed values in fabrication process, thus, any change in the spacings will affect beam-splitting performance. In the following, we analyze the effect of different variations in different cells by comparing the simulation results for 4 kinds of trimer-based metasurfaces: Metasurface 1 and Metasurface 2 are for the case of symmetric near-field coupling, while Metasurface 3 and Metasurface 4 are for the case of asymmetric near-field coupling. Except for d1 and d2, other parameters are the same for these metasurfaces.
Case 1: symmetric near-field coupling
Metasurface 1 is the same metasurface discussed in Fig. 8. The structure unit (cell) is a trimer, multiple cells are periodically arranged along the x- and y-directions with periods Lx=1950nm and Ly=550 nm, respectively. The separations (spacings) between the three Si nanocylinders are d1=d2=10 nm, which represents the case of symmetric near-field coupling.
Metasurface 2 is of non-period structure and it is the counterpart to Metasurface 1 after considering fabrication errors. Due to fabrication errors, the actually obtained parameters will departure from the designed values and the metasurface will lose periodicity. For the sake of simplification and saving simulation time, we assume that Metasurface 2 consists of 14×14 different cells, and we only consider fabrication variations in d1 and d2 because they are the two most crucial parameters that determine near-field coupling and thus beam splitting performance. We set a relative high fabrication error of 30%. In other words, in comparison with the designed values (d1=d2=10 nm) in Metasurface 1, now d1 in different cells in Metasurface 2 varies randomly from 7 nm to 13 nm, corresponding to a random fabrication error (variation) of 30%. Similarly, d2 in different cells also ranges randomly from 7 nm to 13 nm. In the simulation, period boundary conditions are no longer used in this case, instead we endow d1 and d2 with random values within the range of 7 nm∼13 nm by using random number generation function.
For Metasurface 1, Fig. 14(a) shows its transmittance spectra for different orders and the total transmittance spectrum. As can be seen, m= ±1 orders always have equal transmittances in beam-splitting waveband (1141 nm∼1238 nm and 1309 nm∼1436 nm, which are marked by pink areas), meaning Metasurface 1 works as a 1:2 beam splitter as designed.
For Metasurface 2, Fig. 14(b) shows that m= ±1 orders have nearly equal transmittances in beam-splitting waveband (1129 nm∼1230 nm and 1309 nm∼1402 nm, which are marked by the pink areas), meaning that Metasurface 2 can still work as a 1:2 beam splitter despite the 30% random fabrication variations. However, in comparison with Metasurface 1, Metasurface 2 has a narrower operation waveband and smaller transmittances for m= ±1 orders. For Metasurface 1, the maximal transmittance for m= ±1 orders is 38.7% and beam-splitting bandwidth is 224 nm. In contrast, for Metasurface 2, the maximal transmittance for m= ±1 orders is 36.4% and the beam-splitting bandwidth is 194 nm.
Case 2: asymmetric near-field coupling
Like Metasurface 1, Metasurface 3 is of periodic structure, but d1 and d2 are not equal to each other. Instead, d1=10 nm and d2=20 nm, this represents the case of asymmetric near-field coupling.
Metasurface 4 is of non-period structure, it is the counterpart to Metasurface 3 after considering a fabrication error of ±3 nm in d1 and d2. In other words, in comparison with the designed values of Metasurface 3 (d1=10 nm, d2=20 nm), now d1 in different cells for Metasurface 4 ranges randomly from 7 nm to 13 nm, and d2 in different cells ranges randomly from 17 nm to 23 nm.
For Metasurface 3, Fig. 15(a) shows that m= -1 order always has a higher transmittance than m= +1 order in the beam-splitting waveband (1101-1184 nm and 1306-1420 nm, which are marked by pink areas), and the maximal splitting ratio is 1.83.
For non-periodic Metasurface 4, Fig. 15(b) shows that m = -1 order always has a higher transmittance than m= +1 order in the beam-splitting waveband (1103-1186 nm and 1306-1413 nm, which are marked by the pink areas). However, the maximal splitting radio is 2.01, larger than that of Metasurface 3; while the maximal transmittance of m= -1 order is 36.6% for Metasurface 3, and 35.4% for Metasurface 4.
National Natural Science Foundation of China (61675074).
The authors declare no conflicts of interest.
1. J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, “Dual-function beam splitter of a subwavelength fused-silica grating,” Appl. Opt. 48(14), 2697–2701 (2009). [CrossRef]
2. B. Wang, L. Chen, L. Lei, and J. Zhou, “Two-layer dielectric grating for polarization-selective beam splitter with improved efficiency and bandwidth,” Opt. Commun. 294(1), 329–332 (2013). [CrossRef]
3. W. Zhu, B. Wang, C. Gao, K. Wen, Z. Meng, Q. Wang, X. Xing, L. Chen, L. Lei, and J. Zhou, “Dual-function splitting of the embedded grating with connecting layer,” Mod. Phys. Lett. B 33(11), 1850129 (2019). [CrossRef]
4. D. C. Tee, T. Kambayashi, S. R. Sandoghchi, N. Tamchek, and F. R. Mahamd Adikan, “Efficient, wide angle, structure tuned 1×3 photonic crystal power splitter at 1550 nm for triple play applications,” J. Lightwave Technol. 30(17), 2818–2823 (2012). [CrossRef]
5. Y. Zhang, M. A. Al-Mumin, H. Liu, C. Xu, L. Zhang, P. L. Kam, and G. Li, “An integrated few-mode power splitter based on multimode interference,” J. Lightwave Technol. 37(13), 3000–3008 (2019). [CrossRef]
6. X. Chen, W. Liu, Y. Zhang, and Y. Shi, “Polarization-insensitive broadband 2 × 2 3 dB power splitter based on silicon-bent directional couplers,” Opt. Lett. 42(19), 3738–3740 (2017). [CrossRef]
7. D. Zhang, M. Ren, W. Wu, N. Gao, X. Yu, W. Cai, X. Zhang, and J. Xu, “Nanoscale beam splitters based on gradient metasurfaces,” Opt. Lett. 43(2), 267–270 (2018). [CrossRef]
8. A. Ozer, N. Yilmaz, H. Kocer, and H. Kurt, “Polarization-insensitive beam splitters using all-dielectric phase gradient metasurfaces at visible wavelengths,” Opt. Lett. 43(18), 4350–4353 (2018). [CrossRef]
9. M. Wei, Q. Xu, Q. Wang, X. Zhang, Y. Li, J. Gu, Z. Tian, X. Zhang, J. Han, and W. Zhang, “Broadband non-polarizing terahertz beam splitters with variable split ratio,” Appl. Phys. Lett. 111(7), 071101 (2017). [CrossRef]
10. Z. Yang, R. Jiang, X. Zhuo, Y. Xie, J. Wang, and H. Lin, “Dielectric nanoresonators for light manipulation,” Phys. Rep. 701, 1–50 (2017). [CrossRef]
11. N. J. Halas, S. Lal, W.-S. Chang, S. Link, and P. Nordlander, “Plasmons in Strongly Coupled Metallic Nanostructures,” Chem. Rev. 111(6), 3913–3961 (2011). [CrossRef]
12. M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]
13. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016). [CrossRef]
14. S. Kruk and Y. Kivshar, “Functional meta-optics and nanophotonics governed by Mie resonances,” ACS Photonics 4(11), 2638–2649 (2017). [CrossRef]
15. W. Liu and Y. S. Kivshar, “Multipolar interference effects in nanophotonics,” Phil. Trans. R. Soc. A 375(2090), 20160317 (2017). [CrossRef]
16. D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015). [CrossRef]
17. B. Hopkins, D. S. Filonov, A. E. Miroshnichenko, F. Monticone, A. Alu, and Y. S. Kivshar, “Interplay of Magnetic Responses in All-Dielectric Oligomers to Realize Magnetic Fano Resonances,” ACS Photonics 2(6), 724–729 (2015). [CrossRef]
18. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5(1), 5753 (2014). [CrossRef]
19. N. A. Nemkov, I. V. Stenishchev, and A. A. Basharin, “Nontrivial nonradiating all dielectric anapole,” Sci. Rep. 7(1), 1064 (2017). [CrossRef]
20. P. Albella, T. Shibanuma, and S. A. Maier, “Switchable directional scattering of electromagnetic radiation with subwavelength asymmetric silicon dimers,” Sci. Rep. 5(1), 18322 (2016). [CrossRef]
21. Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-resonant dielectric metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015). [CrossRef]
22. R. M. Bakker, D. Permyakov, Y. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and Electric Hotspots with Silicon Nanodimers,” Nano Lett. 15(3), 2137–2142 (2015). [CrossRef]
23. J. Ding, Y. Ling, J. Luan, W. Wu, S. Li, and L. Huang, “High-efficiency and Broadband Optical Beam Splitters Based on All-dielectric Polymer-based Metasurfaces,” 2018 Asia Communications and Photonics Conference (ACP), Hangzhou, 2018, pp. 1–3, doi: 10.1109/ACP.2018.8595739.
24. E. Khaidarov, H. Hao, R. Paniagua-Domínguez, Y. T. Toh, J. S. K. Ng, and A. I. Kuznetsov, “Asymmetric nanoantennas for ultra-high angle broadband visible light bending,” Nano Lett. 17(10), 6267–6272 (2017). [CrossRef]
25. R. Paniagua-Dominguez, Y. Yu, E. Khaidarow, S. Choi, V. Leong R, M. Bakker, X. Liang, Y. Fu, V. Valuckas, L. A. Krivitsky, and A. I. Kuznetsov, “A Metalens with Near-Unity Numerical Aperture,” Nano Lett. 18(3), 2124–2132 (2018). [CrossRef]
26. P. R. Wiecha, A. Cuche, A. Arbouet, C. Girard, G. Francs, A. Lecestre, G. Larrieu, F. Fournel, V. Larrey, T. Baron, and V. Paillard, “Strongly Directional Scattering from Dielectric Nanowires,” ACS Photonics 4(8), 2036–2046 (2017). [CrossRef]
27. J. Tian, Q. Li, Y. Yang, and M. Qiu, “Tailoring Unidirectional Angular Radiation through Multipolar Interference in a Single-element Subwavelength All-Dielectric Stair-like Nanoantenna,” Nanoscale 8(7), 4047–4053 (2016). [CrossRef]
28. M. Panmai, J. Xiang, Z. Sun, Y. Peng, H. Liu, and S. Lan, “All-silicon-based nano-antennas for wavelength and polarization demultiplexing,” Opt. Express 26(10), 12344–12362 (2018). [CrossRef]
29. I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring Directional Scattering through Magnetic and Electric Resonances in Subwavelength Silicon Nanodisks,” ACS Nano 7(9), 7824–7832 (2013). [CrossRef]
30. M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “High-Efficiency All-Dielectric Metasurfaces for Ultra-Compact Beam Manipulation in Transmission Mode,” Nano Lett. 18(12), 7801–7808 (2015). [CrossRef]
31. J. van de Groep and A. Polman, “Designing dielectric resonators on substrates: Combining magnetic and electric resonances,” Opt. Express 21(22), 26285–26302 (2013). [CrossRef]
32. D. Vercruysse, Y. Sonnefraud, N. Verellen, F. B. Fuchs, G. Martino, L. Lagae, V. V. Moshchalkov, S. A. Maier, and P. van Dorpe, “Unidirectional Side Scattering of Light by a Single-Element Nanoantenna,” Nano Lett. 13(8), 3843–3849 (2013). [CrossRef]
33. J. Li, N. Verellen, D. Vercruysse, T. Bearda, L. lagae, and P. Dorpe, “All-Dielectric Antenna Wavelength Router with Bidirectional Scattering of Visible Light,” Nano Lett. 16(7), 4396–4403 (2016). [CrossRef]