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Abstract
Metasurfaces have been widely investigated for their capabilities of manipulating wavefront versatilely and miniaturizing traditional optical elements into ultrathin devices. In this study, a nanoscale tunable beam splitter utilizing a bilayer of geometric metasurfaces in the visible spectrum is proposed and numerically examined. Inspired by the diffractive Alvarez lens and multilayer geometric metasurfaces, opposite quadratic phase distributions are imparted on both layers, and a varying linear phase gradient will arise through relatively lateral displacement between two layers, generating tunable angles of deflection. In addition, such geometric metasurfaces offer opposite directions of phase gradients for orthogonal circularly polarized incidences, leading to effective polarization beam splitting. Results prove that the splitting angles can be tuned precisely, and the energy split ratio can be effectively changed according to the ellipticity of the polarized incidence. This design could find significant applications in optical communication, measurement, display, and so on.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A beam splitter is an optical component that splits one incident light into two or more beams, and it plays a crucial part in optical and photonic applications, such as interferometers, optical communications and display systems. Traditional beam splitters, which transmit half of the incidence light and reflect another half, can be constructed by pasting two triangular prisms together to make a cube splitter, or by a flat plate with thin coatings on one or both surfaces. Further methods and configurations are utilized to implement various beam splitters, for example, gratings [1–3], wave plates [4–6], photonic crystals [7–9], metasurfaces [10–16], and so on. Different types of beam splitters are able to control one or more properties of the output beams, such as polarization states, splitting angles, energy split ratio and wavelength separation.
Metasurfaces, ultrathin nanostructures composed of subwavelength dielectric or metallic antennas [17], have drawn extensive attention in their extraordinary wavefront shaping capability for enormous applications, including meta-lenses [18–22], holograms [23–26], vortex beam generators [27–29] and so on, and beam splitters implemented by metasurfaces are very advantageous in being integrated into miniature optical circuits, for instances, on-chip optical interferometers, multiplexers and demultiplexers. Furthermore, the tuning of functionalities over a wide and continuous range is of substantial interest in metasurface-based beam splitters (MBSs), and bilayer metasurfaces inspired by Alvarez lens design [30–32] are very feasible for adjusting their splitting angles and other properties. Such device changes the total phase distribution by relative displacement between both layers and perform more accurate tuning abilities than stretchable metasurfaces on flexible substrates [33–35], microelectromechanical systems (MEMS)-based metasurfaces [36] and other active metasurfaces. In addition, by operating independently on the left circular polarization (LCP) component and right circular polarization (RCP) components decomposed from the incidence, geometric metasurfaces (GEMs) offer possibilities of controlling the intensity split ratios of the output beams with orthogonal circular polarizations [37].
In this paper, we propose a bilayer geometric metasurfaces based beam splitter (GMBS) composed of titanium dioxide (TiO2) nanofins on quartz substrates, which can continuously and precisely tune the splitting angles and split ratios in the entire visible spectrum. On one hand, through constructing both layers of the bilayer GMBS with opposite quadratic phase distributions and actuating lateral movement between them, a changing phase gradient will be generated in the total phase distribution for tuning the splitting angles. On the other hand, the bilayer GMBS will impart the orthogonal LCP and RCP components with opposite phase distributions independently by geometric phase modulation [38], directing the output LCP beam and RCP beam at opposite splitting angles. In addition, the split ratio of both beams is tuned in accordance with the polarization state of the incidence, which can be arbitrarily polarized from circular polarization (CP) through elliptical polarization (EP) to linear polarization (LP). The characteristics of the bilayer GMBS are theoretically investigated and numerically verified by finite difference time domain (FDTD) method, and applications of this device are envisioned in particular for optical communications, measurement, display.
2. Theoretical analyses
The basic working principles of the bilayer GMBS are described as follows. Metasurfaces can manipulate the directions of the transmitted waves in accordance with the generalized Snell’s law of refraction [39]
Tunable angle of refraction is then achieved by altering the phase gradient via a two-step process: first, construct both layers of metasurfaces with proper phase distributions; second, change the total phase of the bilayer metasurfaces by laterally mechanical motions between both layers. ϕGEM1(x) = ax2 and ϕGEM2(x) = -ax2 are used to express the phase distributions of the lower layer GEM1 and the upper layer GEM2 of the bilayer metasurfaces as shown in Fig. 1, where x represents the horizontal position along metasurface planes in the Cartesian coordinates and a is a constant representing the rate of phase variation. If the lower layer is moved laterally by a distance d along the x axis, the total phase distribution of the bilayer metasurfaces ϕT(x) will be
The final phase gradient dϕT(x)/dx then becomes -2ad, which has a form of linear variation of the displacement value d. When an appropriate value of a is applied and a normally incident incoming light hits the metasurface, the transmitted beam will be deflected at an angle of
Fig. 1. Design of the bilayer GMBS. The nanofins of both GEM layers are arranged face to face with a small gap distance g. An arbitrarily polarized light normally hits the device and splits into one LCP beam and one RCP beam in opposite directions. The splitting angles θL and θR can be tuned continuously by laterally moving the bottom layer GEM1 with a displacement d. The split ratio changes when the polarization state of the incidence alters. The inset depicts the schematic of the unit cells of both GEM layers.
According to Eq. (3), the direction of the deflection can be tuned by the lateral displacement of the lower layer GEM1. However, the incoming light will be bent towards only one direction since the total phase gradient is constant. Only if the total phase gradient owns different values for the whole incident light [13] or for different parts of the light, the transmitted light will be split into beams with different angles. A possible way to attain that goal is to decompose the incident light into orthogonal polarizations and impart both with different phase gradients by one interface. GEMs operate for circularly polarized light and manipulate the local phase by rotating nanofins of metasurface unit cells [37], which imparts opposite phase distributions for LCP and RCP incidences. In detail, the GEMs work similarly to half-wave plates, and when a circularly polarized beam is incident on the nanofin, the transmitted light can be described by
3. Results and discussions
3.1 Tunability of beam splitting angles
In this design, titanium dioxide (TiO2) nanofins, which have high refractive index and transmittance in the visible range [40], are placed on a SiO2 substrate and used to modulate the phase of incident light, as shown in the inset of Fig. 1. Each TiO2 nanofin has a high aspect ratio and an elliptical cross section, with longer length DL, shorter length DS, height h and periodic spacing p set to be 228 nm, 103 nm, 600 nm and 250 nm, respectively. As shown in Fig. 2, these structural parameters ensure that the polarization conversion efficiency (PCE) of the nanofin is relatively high across the visible range, which is a prerequisite for efficient and broadband wavefront manipulation of the bilayer GMBS. The dips in Fig. 2 are produced from the resonance of TiO2 nanofins and cannot be avoided when a minimum PCE value of 20% is required during structural parameter sweeping.
Fig. 2. Simulated polarization conversion efficiency (PCE) of the designed nanofin. PCE represents the proportion of the CP incidence that is converted to transmitted light with opposite helicity of polarization state, and is irrelevant to the rotation angle of the nanofin.
In practical manufacturing, maintaining high PCE across the entire visible spectrum requires extremely high fabrication precision, but typical top-down etching techniques for implementing dielectric metasurfaces would introduce significant surface roughness and sidewalls that are not vertical in TiO2 nanofins [41], which are unacceptable for TiO2 metasurfaces. As a solution, a bottom-up nanofabrication via atomic layer deposition (ALD) should be employed in making metasurfaces composed of amorphous TiO2 nanofins in order to enable high-aspect-ratio structures and low surface roughness [18,41–43]. The standard process starts with creating nanofin patterns by electron-beam lithography (EBL) in the electron beam resist after spin-coating the fused silica substrate with the resist as thick as the height of the designed nanofin, and the remaining resist should form hole-like structures that are exactly opposed to the final structures. Next, ALD should be used to fill amorphous TiO2 into the holes, and TiO2 should exist not only in structure but also on the top of resist with the overcoated deposition thickness of at least half width of the nanofin for producing void-free nanofins. As a key step, the overcoated TiO2 film on top of the resist should be of equal thickness [18], which is the most challenging and difficult part during the actual fabrication operation. After finishing ALD, the inductively coupled plasma reactive ion etching (ICP-RIE) is assumed to fully remove the overcoated TiO2 layer so that the resist will be exposed. Finally, high-aspect ratio and anisotropic TiO2 nanostructures with minimal surface roughness will remain after stripping the remaining electron beam resist.
A value of π/3 (rad/µm2) is chosen to be the coefficient of phase variation rate a. The size of each metasurface layer is set as -10 µm < x < 10 µm, and this device is assumed to extend to infinity in y-direction as it works like a one-dimensional grating. Simulations are then carried out by using the commercial electromagnetic simulation software package (FDTD Solutions), where perfect matching layer (PML) boundary condition is applied at x-direction and z-direction while periodic boundary condition is applied at y-direction. As for selecting an appropriate gap distance between two metasurface layers for realizing the superposition of ϕGEM1 and ϕGEM2, half of Talbot distance which is defined as 2p2/λ would be the optimal distance for making the desired total phase that combines phases of both layers of metasurfaces [44–46]. However, this value ranges from 89.3 nm to 156.3 nm, which is inappropriate to choose since it is not only too small for axial alignment as well as practical applications, but also likely to introduce non-negligible near-field effect. Hence, we select 800 nm as the gap distance, which is a multiple of the Talbot distance and a little larger than the incident wavelength range. In addition, this distance is small enough to avoid significant diffraction of the output wavefront of GEM1 before it hit GEM2, as shown in Fig. 3(a) that the simulated phase distributions are very close to theoretical predictions. The mean value of the difference between the total phase ϕT and the sum of phase profiles of both layer ϕGEM1+ϕGEM2, which is ϕdiff = |ϕT-(ϕGEM1+ϕGEM2)|, is only less than π/5, indicating that the superposition of both phases is working for the design.
Fig. 3. (a) Phase profiles of ϕGEM1, ϕGEM2 and their superposition ϕT after GEM1 is moved laterally by a distance of 3.99 µm along x axis for splitting angle of 45°. The dashed lines represent the designed phase profiles and the solid lines represent the simulated phase profiles. (b)-(d) Simulated phase distributions from output electric fields of GEM1 under LCP incidence, GEM2 under RCP incidence and the bilayer GMBS under LCP incidence, respectively, corresponding to the actual phase profiles in (a).
In the following discussion, the performance of the tunable splitting angles is analyzed, and the characteristic of the continuously changed split ratio of both split beams will be investigated. In order to demonstrate how phase distributions of both metasurface layers influence on the total phase, a case of d = 3.99 µm in which the splitting angle is expected to be 45° in terms of Eq. (3) is firstly simulated. In this case, LCP incidence normally hits GEM1 and is converted to RCP light, which is then converted to LCP light again after passing through GEM2. Figure 3 exhibits phase profiles of ϕGEM1 and ϕGEM2 by hitting each of the two metasurface layers with LCP and RCP plane waves, separately, and total phase distribution is obtained by an identical LCP plane wave hitting GEM1. It can be clearly seen that the simulated phase profiles are very close to theory predictions, and the simulated total phase profile follows a linear phase gradient with a slope of -8.37 (rad/µm), which generates a deflection angle θL = -45°. Various values of displacement d and LP incidence (x-polarized) are then applied in the simulations to explore the beam splitting characteristics. The real components of the extracted LCP and RCP in the output electric field under 532nm incidence, corresponding to d = 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm, are depicted in Fig. 4(a-b, d-e, g-h and j-k), and their calculated far field intensity distributions are clearly observed and shown in Fig. 4(c, f, i and l). In the polar coordinates, the dot points represent the diffraction efficiencies of orders with various diffraction angles, and they clearly illustrate that the maximum intensities appear at the angles of ±15°, ±30°, ±45° and ±60°. It can also be seen from Fig. 4 that LP incident will be split equally, and the splitting angles increase with the displacement augmenting.
Fig. 4. (a, d, g and j) The extracted transmitted LCP electric field distributions (real part) under normal incidence of x-polarized light corresponding to d = 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm, respectively. (b, e, h and k) The extracted transmitted RCP electric field distributions (real part) under normal incidence of x-polarized light corresponding to d = 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm, respectively. (c, f, i and l) Calculated far-field radiation patterns under normal incidence of x-polarized light corresponding to d = 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm, respectively.
In addition, more data of different incident wavelengths and various displacement values are simulated and demonstrated in Fig. 5 to support that conclusion. Figure 5(a) shows that the splitting angles depend on the displacement of GEM1 as well as the incident wavelengths. As the wavelength of the incident light and displacement value increase, the splitting angle augments. By comparing our simulated results (symbols) corresponding to 473nm, 532nm, 593nm and 633nm with the theoretical curves (dashed line) calculated according to Eq. (3), it is confirmed that the simulation results agree well with the theoretical predictions.
Fig. 5. (a) Splitting angle dependent on the lateral displacement of GEM1 with incident wavelengths of 473nm, 532nm, 593nm and 633nm. The dashed lines are theory predictions according to Eq. (3) and the symbols represent the simulated results. (b) Diffraction efficiency as a function of incident wavelength corresponding to d = 1 µm, 2 µm, 3 µm, 4 µm and 5 µm, respectively. Diffraction efficiency is defined as the summed power of the output LCP and RCP transmitting in the desired splitting angles to the incident power with arbitrarily polarization state.
For the designed device, diffraction efficiencies are defined as the summed power of the transmitting LCP and RCP waves to the total incident power, which are calculated and depicted in Fig. 5(b) with the value of displacement altering from 1 µm to 5 µm. It can be found out that the efficiency deceases with the as the splitting angle (or the displacement value) increases, and the device is supposed to obtain the highest efficiency when d = 0 as no phase gradient is imparted onto the incident wave. Moreover, the diffraction efficiency may degrade at large deflection angles. When the desired splitting angle reaches near 70°, the diffraction efficiencies decrease dramatically, and the majority power of the transmitted field focuses in the non-split power of the zero order as demonstrated in Eq. (8). It is worth noting that efficiencies of identical displacement values keep consistent with PCE of the designed nanofin in Fig. 2.
3.2 Split ratio tuned by ellipticity of polarized incidence
The previous simulations are carried out under linearly polarized light, resulting in identical split ratio, which is defined as the power ratio of the split LCP and RCP beams. While it is highly desirable to achieve various split ratios for the bilayer GMBS to be applied in diverse optical systems. There are two orthogonal cross-polarized parts, LCP and RCP, in the transmission field when the metasurfaces are normally illuminated by the EP light in terms of Eq. (5), and for practical applications, two different split plane waves with arbitrarily relative intensities can be obtained by rotating the quarter-wave plate in front of the bilayer GMBS. Based on Eq. (8), two propagating waves with opposite polarization states appear simultaneously in the far field, and it is evident that adjusting the ellipticity of incident light, which is defined as η = (IRCP-ILCP)/(IRCP+ILCP), will alter the split ratio between both output beams. IRCP and ILCP correspond to the intensities of the output RCP and LCP beams, η = ±1 represents RCP/LCP and η = 0 represents LP. By changing the ellipticity of incident light, the polarization state can be tuned continuously from LCP/RCP, through EP and LP, to RCP/LCP, leading to different relative intensities of the splitting beams.
Then simulations are carried out with incident wavelength of 532nm and the aimed splitting angles fixed at ±30° when d = 2.82 µm. The relationship between the relative intensity of the two splitting beams and ellipticity of the incident light is investigated and shown in Fig. 6, which confirms that the split ratio can be controlled by the ellipticity of the incidence. The diffraction efficiencies of the LCP and RCP splitting beams and their summed values under incidence of various ellipticity are described in Fig. 6(a), and their relative intensities are depicted in Fig. 6(b) as well. As η continuously increases from -1 to 1, the relative intensity of two splitting beams gradually changes from 1:0, through 0.5:0.5 to 0:1, which fit the theory predictions of Eq. (8) well. The simulated results prove that the split ratio of the bilayer GMBS can be conveniently and flexibly modulated by changing the ellipticity of incidence without redesigning another phase profile of each layer, and it is reasonable to expect that it is highly stable and insensitive to the changes of wavelength, energy loss during light propagation and diffraction, and structural parameters.
Fig. 6. (a) Simulated diffraction efficiencies of LCP output beam, RCP output beam and the sum of them under normal polarized incidence with ellipticity ranging from -1 to 1. (b) Relative intensities of LCP and RCP under incidence with different ellipticity. Dashed lines represent the theory predictions and symbols (square and circle) means the simulated results.
In order to comprehensively illustrate the beam splitting characteristics of the bilayer GMBS, far-field diffraction intensity distributions with different displacement values and various polarization states of incidence are displayed in Fig. 7. Here, cases of d = 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm are investigated, under polarized 532nm incidence of η = -1, -0.5, 0.5 and 1 rather than η = 0 in Fig. 4. When η = ±1, pure RCP or LCP incidence will hit the device and the output beam will be directed at one deflection angle with only RCP or LCP polarization state; In addition, upon incidences with η = ±0.5, the decomposed RCP beam and LCP beam from the incident light should have a relative intensity (split ratio) of 3:1 or 1:3. Figure 7 clearly shows that the rough diffraction efficiency contrasts of the LCP and RCP beams are close to 1:0, 3:1, 1:3 and 0:1, respectively, regardless of the splitting angles. This accords accurately with the predicted results from Eq. (3) and Eq. (8), demonstrating that the splitting angles and split ratio are independently and precisely tuned by the lateral displacement of GEM1 and the ellipticity of incidence, respectively.
Fig. 7. Calculated far-field radiation patterns of various splitting angles under 532nm incidence of different ellipticity. (a, b, c and d) Calculated far-field radiation patterns corresponding to η = -1, -0.5, 0.5 and 1, respectively, with d = 1.46 µm for splitting angles of ±15°. (e, f, g and h) Calculated far-field radiation patterns corresponding to η = -1, -0.5, 0.5 and 1, respectively, with d = 2.82 µm for splitting angles of ±30°. (i, j, k and l) Calculated far-field radiation patterns corresponding to η = -1, -0.5, 0.5 and 1, respectively, with d = 3.99 µm for splitting angles of ±45°. (m, n, o and p) Calculated far-field radiation patterns corresponding to η = -1, -0.5, 0.5 and 1, respectively, with d = 4.88 µm for splitting angles of ±60°.
Furthermore, this bilayer GMBS of larger dimensions, which would make more sense in practice, should be discussed, since the proposed design has a very small lateral dimension smaller than 20 µm and is not realistic for real applications. The results from the full-wave simulations by FDTD already proves the accuracy as well as the effectiveness of geometric phase modulation of the bilayer metasurfaces, and metasurfaces provide very frequent phase sampling at sub-wavelength scales, therefore the splitting performance can be further illustrated by Rayleigh–Sommerfeld’s scalar diffraction theory for larger areas. It can be observed from Fig. 8 that when the size is enlarged from 20 µm to 0.6 mm, the beam splitting characteristic behaves as good as predicted by the far-field patterns in Fig. 5, and proves again the ignorable effect of diffraction when the gap distance is selected as 800 nm.
Fig. 8. Beam splitting characteristic simulated by scalar diffraction theory when the lateral dimension of the bilayer GMBS is enlarged to 0.6 mm . The incidence is 532 nm x-polarized light and the gap distance is fixed at 800 nm. (a, b, c and d) GEM1 is considered to be moved laterally by a distance of 1.46 µm, 2.82 µm, 3.99 µm and 4.88 µm, respectively, for splitting angles of ±15°, ±30°, ±45° and ±60°, respectively.
Last but not least, this design is supposed to be applied in a good translation stage with high resolution to activate the tunability, thus how the alignment and tunability can realistically be achieved in practice should also be discussed. The ideal experimental setup and alignment are proposed and shown in Fig. 9, where the tunable behavior of the designed bilayer GMBS should be experimentally verified by laterally displacing GEM1 with respect to GEM2. After fixing the position of GEM2, GEM1 is designed to be laterally actuated by an extremely precise nanopositioning system as the translation stage. The translation stage should have a minimum motion resolution less than 10 nm and a travel range larger than 5 µm in this design, and a number of nanopositioning systems could meet these requirements. For example, P-545.3R8S XYZ Piezo System from Physik Instrumente offers a closed loop resolution of 1 nm and a travel range of 200 µm in three dimensions. In addition, elliptically polarized light will be generated by the relative rotation between the linear polarizer and the quarter wave retardation plate, and the LCP and RCP components will be separated after the incident beam passes the bilayer GMBS. Finally, a large-area rotating breadboard is utilized for receiving transmitted beams from different angles by a photodetector.
Fig. 9. Schematic of the proposed experimental setup used for measuring the diffraction patterns of the enlarged bilayer GMBS. The laser beam could firstly be collimated by a fiber collimator with a beam size diameter about 0.6 mm, and then the beam would pass through a fixed linear polarizer and a rotating quarter-waveplate to generate elliptically polarized light. Ideally, GEM1 is fixed on a nanopositioning system which is able to actuate the lateral displacement of GEM1 with a resolution less than 10 nm, while GEM2 is clamped on a fixed stage. The beam should be split into two beams with LCP and RCP components, respectively, and received by a photodetector for recording the intensity distribution after an objective and a tube lens. The beam receiving parts are fixed on a motor-controlled stage which rotates with a large-area rotating breadboard in order to receive split beams from different angles.
4. Conclusions
In conclusion, a beam splitter made by a bilayer of metasurfaces working in the entire visible spectrum with tunable splitting angles and split ratios has been proposed. The principle of the beam splitter is based on the generalized Snell’s law and geometric phase modulation method, and cascaded bilayer metasurfaces comprising elliptical titanium dioxide nanofins are used in this design. Based on this idea, the splitting angles and split ratio of the designed beam splitter are verified to be effectively tunable. On the one hand, the splitting angle is precisely tuned by laterally changing the relative displacement between both layers. On the other hand, the split ratio is continuously changed by the ellipticity of the polarized incidence. This bilayer GMBS with flexible tunability is hopeful to find wide applications in compact photonic devices, such as miniature interferometers, multiplexers for integrated optical circuits, optical communication systems, augmented reality display and so on.
Funding
National Natural Science Foundation of China (61327902).
Disclosures
The authors declare no conflicts of interest.
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