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Abstract
We propose a flat retroreflector that can efficiently reflect the electromagnetic waves back along its incident direction in a wide continuous range of angles. This retroreflector consists of a quadratic metalens and a flat metallic reflector at the focal plane of the former. The quadratic metalens is a dielectric pillar array encoded with a quadratic phase profile and it is embedded in the top side of the substrate. The flat reflector is on the bottom side of the substrate. The designed retroreflector has a diameter of 40 mm, a thickness of 15 mm, and a working frequency of 77 GHz. Through meta-units optimization, a retroreflection efficiency of 38.51% at ± 60° incidence and an average retroreflection efficiency of 46.39% for the incident angles from 0° to 60° can be numerically demonstrated. This flat retroreflector is easy for integration, which is promising for potential applications in the miniature wireless communication systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Retroreflector is a device that can reflect the electromagnetic (EM) wave back along its incident direction. Conventional retroreflectors, including Luneburg lens reflector [1,2], corner cube reflector [3,4] and cat's eye reflector [5,6], can realize retroreflection over a continuous range of incident angles. They have been widely used for applications in the range from microwave to optical wavelengths, such as radar cross-section augmentation [2], free-space communication [3–5] and length measurements [6]. However, the structures of these conventional retroreflectors (spherical or cubiform bases) limit their applications in aircrafts, missiles, or miniature wireless communication systems. A flat device is easier for integration and installation, which is desirable for these applications. A kind of flat retroreflector named Van Atta retroreflector was proposed in 1959 [7]. It realizes retroreflection using antenna array in which antennas are connected in symmetrical pairs by transmission lines of equal length. The Van Atta retroreflector has been widely studied from microwave to terahertz bands for target tracking and wireless communication [8,9]. However, the distribution of transmission lines become complicated and the ohmic loss is increased when the number of antennas increases. Recently, flat retroreflector based on metasurfaces has attracted extensive attentions because metasurface has the capabilities to manipulate the phase, amplitude and polarization of EM waves at a subwavelength scale. Some early works based on the phase gradient metasurfaces (PGMs) have been demonstrated in the microwave band [10,11]. But those retroreflectors can only operate at a single incident angle, which is limited intrinsically by the uniform phase gradient. Although Maochang Feng et al. [12] increase the number of operating angles (i.e., the angle at which can realize retroreflection) to four by dividing the PGMs into four parts with different phase gradients, the range of operating angle is still discrete.
To realize retroreflection in a continuous range of operating angle, Arbabi et al. demonstrate a retroreflector, which includes a metalens and a reflective metasurface [13]. Their devices have an operating angle ranging from −25° to 25° and a retroreflection efficiency of 50%. The metalens focuses the incident light on the reflective metasurface at different locations for different incident angles. The focused light is reflected by the reflective metasurface encoded with varying phase gradients. The retroreflector has a total diameter of 600 µm and only responses to a light beam with diameter of 34 µm. To enlarge the working aperture, Yong Qiang Liu et al. [14] use a hyperbolic metalens and a curved mirror to realize the retroreflector. A plane wave is focused on the curved mirror by the metalens and produces specular reflections. The retroreflection happens when the incident rays and the corresponding reflected rays after the metalens distribute symmetrically along the chief ray. Moreover, the chief rays under different incident angles should always be the normal of the reflector. The specular reflection on a mirror does not introduce additional diffraction or scattering. The designed retroreflectors have an operating angle ranging from -30° to 30° and a retroreflection efficiency of 50% efficiency. However, the efficiency decreases below 5% when the incident angle is over ± 40°.
To further enlarge the range of operating angles, we introduce a high numerical aperture (NA) quadratic metalens [15,16] into the retroreflector. The high NA quadratic metalens can convert the rotational effect of an oblique incident beam to the translational symmetry of the focusing beam [15], which enlarges the range of operating angles correspondingly. In this letter, we numerically demonstrate a flat retroreflector that has an operating angle ranging from -60° to 60°. In addition, we improve its retroreflection efficiency through the meta-units optimization in terms of period, height, and refractive index.
2. Theory and design
Because the quadratic metalens can convert the motion of EM waves from rotation to translation [15], the focusing beam is symmetrical about the chief ray for different incident angles, and the chief ray is always perpendicular to the focal plane. After a specular reflection, the reflected beam has the same shape as the focusing beam but in reverse propagating directions. As shown in Fig. 1, for incident EM waves with different incident angles, the EM fields after the metalens have different translational distances along the x-axis while the phase profiles are unchanged about focal points. As a result, the phase difference between an arbitrary point A’ and the center point M’ after the metalens under normal incidence (δA’M’) is equal to the phase difference between the corresponding point B’ and N’ under oblique incidence (δB’N’). Point M(M’) is at the center of the metalens and is also at the position of x = 0. Point A(A’) is at the position of x. Assume that φL(x) represents the phase delay introduced by the quadratic metalens at x. The phase difference between B and N before the metalens is induced by the oblique incidence angle of θ, which is equal to n1k0xsinθ. The required phase relationship can be expressed as below:
Fig. 1. (a) The 3D view and (b) the cross-sectional view of the schematic of the flat retroreflector based on the quadratic metalens. The inset in (a) shows the meta-unit with radius R, height H, and period P.
The phase profile of the quadratic metalens φL(x) can be derived from Eq. (1):
The phase profile has the same expression as it in Ref. [15]. It is pivotal to realizing retroreflection under a continuous range of incident angles with a flat configuration whereby the quadratic metalens can keep the shape of the focusing beam unchanged about the chief ray.
For effective EM wave propagation, the condition $|{{{n}_\textrm{2}}{{k}_\textrm{0}}\mathrm{sin\theta }} |\le\mathrm{2\pi }{{n}_\textrm{2}}\textrm{/}{\mathrm{\lambda }_\textrm{0}}$ must be satisfied, or else it will lead to an evanescent wave propagating along the surface of metalens instead of contributing to the focal spot. Considering the conditions of wavevector conservation and Eq. (3), the diameter of an effective aperture Deff can be expressed as: [17]
which means that only the EM wave inside the effective aperture can transmit through the quadratic metalens and contribute to the focal point even if the whole metalens is illuminated.The field of view (FOV) of the quadratic metalens correlates with the NA and is nearly 180° when NA ≥ 0.8 [16]. NA is defined as D/(D2+4F2)1/2 where D is the diameter of the metalens. Because the quadratic metalens with high NA promises a wide range of operating angles, we adopt a quadratic metalens with an NA of 0.8 for the designed retroreflector.
In our design, the diameter of the retroreflector is 40 mm and the thickness is 15 mm. The material of the substrate is resin with a permittivity εr = 2.25 and nearly lossless in millimeter wavebands. The quadratic metalens is a dielectric pillars array, which is embedded in the top side of the substrate, and we firstly use MgO-TiO2 (εr = 16, tanδ = 0.003) pillars in the design. When we change the radius of the pillar, the corresponding relative phase delay is varied from 0 to 2π. To reduce the simulation time and memory burden, a flat PEC layer with a thickness of 0.4 mm is placed at the bottom of the substrate as a reflector. The PEC layer can be replaced by an aluminum layer without changing the reflectivity significantly. The whole device is simulated by the finite-difference time-domain (FDTD) method. To save the simulation time, we use the periodic boundary condition along the y-direction and use the perfectly matched layer boundary condition along the x- and z-directions, i.e., a cylindrical metalens is applied in the retroreflector for the retroreflection efficiency optimization.
We calculate the retroreflection efficiency ηr of the device within the effective aperture based on the simulation results. The retroreflection efficiency ηr can be expressed as ηr = Rηf, where R is the reflectivity of retroreflector (the ratio of the reflected power to the incident power) and ηf is diffraction efficiency in the direction of retroreflection. The diffraction efficiency ηf is given as ηf = Pr/Pt, where Pr is the retroreflected power and Pt is the total diffracted power of the reflected EM waves in the far-field within ±5° cone angle at the direction of and other directions, respectively. The retroreflection efficiency with incident angles from 0° to 60° in 10° step is calculated to evaluate the range of operating angle of the retroreflector.
To optimize the performance of the retroreflector, we calculate the retroreflection efficiency ηr of the device consisting of different meta-units. Subsequently, the average of retroreflection efficiency $\overline {{\mathrm{\eta }_\textrm{r}}} $ based on above incident angles are calculated as the evaluation indicator to optimize the meta-units.
3. Results and discussions
Meta-unit as the building block of metalens should have high transmission and can produce phase delay ranging from 0 to 2π. Figures 2(a)-(b) show the transmittance and relative phase delay of the uniform MgO-TiO2 pillar arrays correlated to the height and radius, respectively when the period is 1 mm. The meta-units with parameters at the left side of the black dotted line in Fig. 2(a) can fully cover the phase range from 0 to 2π as shown in Fig. 2(b). The required radius range is smaller when the pillars get higher.
Fig. 2. (a) The transmittance and (b) the relative phase delay of the uniform MgO-TiO2 pillar arrays (P = 1 mm) embedded in a resin substrate when the incidence is a plane wave with a frequency of 77 GHz. The meta-units with parameters at the left side of the black dotted line can fully cover the phase range from 0 to 2π.
Then, we map the meta-units based on the phase profile of quadratic metalens. Figures 3(a)-(b) show the electric field distributions in the xz-plane of the quadratic metalens under the normal incidence (θ = 0°) and the oblique incidence (θ = 60°) with TM polarization respectively. The meta-units in the metalens have a period of 1 mm, a height of 2.8 mm and a radius ranging from 10 µm to 390 µm.
Fig. 3. The electric field distributions in the xz-plane of the quadratic metalens under (a) the normal incidence and (b) the oblique incidence (θ = 60°); and the electric field distributions in the xz-plane of the retroreflector under (c) the normal incidence and (d) the oblique incidence (θ= 60°) with TM polarization. The white lines show the positions of the quadratic metalens and PEC layer
The specular reflection happens after adding the flat PEC layer. The quaduatic metalens makes the reflected beam having the same shape as the focusing beam but in reverse propagating directions. Figures 3(c)-(d) show the electric field distributions in the xz-plane of the retroreflector under the normal incidence and the oblique incidence (θ = 60°), respectively. Based on the parallel interference fringes and the field distributions, we can conclude that, for the normal incidence, the effective aperture is at the central area of the device, and, for the oblique incidence, it is located at one side.
To calculate the retroreflection efficiency, we need to evaluate the diffraction efficiency of the retroreflector. Based on the near-to-far-field projection method, the far-field intensity distribution is calculated on a hemispherical surface located 1 m above the retroreflector. Figures 4(a)-(d) show the far-field intensity distributions of reflected waves for the incident angle at 0°, 20°, 40°, and 60°, respectively. Most power is reflected along the incident direction for each incidence angle. In addition, the diffraction powers in other directions including the direction of normal reflection are well suppressed. Figure 4(e) shows the far-field intensity distribution under different incident angles from 0° to 70° in 2° step. The main beam of the reflected EM waves does not propagate along the incident direction anymore when the angle of incidence is larger than 62°.
Fig. 4. The far-field power distributions of reflected waves at the incident angle of (a) 0°, (b) 20°, (c) 40°, and (d) 60°, respectively. Red lines show the far-field power distributions at the xz-plane. θin, θre, and θn represent the angle of incidence, retroreflection, and normal reflection, respectively. (e) The far-field intensity distribution under different incident angles from 0° to 70° in 2° step.
We optimize the meta-units and calculate the retroreflection efficiency with different incident angles ranging from 0° to 60° in the 10° step to evaluate the performance of the retroreflector. Figure 5(a) shows the retroreflection efficiency ηr of the device vs the period of meta-units when H = 2.8 mm. Considering the fast changing phase delay at the edge of high NA metalens, the period of meta-units has to be smaller than 1.3 mm to ensure that there are at least two meta-units used to map a segment of phase profile of the metalens from 0 to 2π. As such, we calculate the efficiencies of the designed retroreflectors consisting of meta-units with the period ranging from 0.8 mm to 1.2 mm. When the period of meta-units is equal to 1 mm, the retroreflector has a high efficiency for all incident angles. Figure 5(b) shows the retroreflection efficiency ηr of the device vs the height of meta-units when P = 1 mm. In this condition, the retroreflector has a higher efficiency for all incident angles when the height of meta-unit is 2.8 mm. Considering the average retroreflection efficiencies ηr (from 0° to 60° in 10° step) as shown in Fig. 5(c), it reaches the highest value of 45.63% at the period of 1 mm. We also calculate the average retroreflection efficiencies ηr of the devices with different heights of the meta-unit as shown in Fig. 5(d). The average efficiency reaches the highest value of 45.63% when H = 2.8 mm.
Fig. 5. The retroreflection efficiency ηr of the designed retroreflector under the incident waves with different incident angles when the corresponding meta-units have different (a) periods and (b) heights. The average retroreflection efficiency varies versus (c) the period and (d) the height of meta-units.
To further increase the retroreflection efficiency, we consider the retroreflectors consisting of meta-units with different materials. In Table 1, $\mathrm{\bar{R}}$, $\overline {{\mathrm{\eta }_\textrm{f}}} $ and $\overline {{\mathrm{\eta }_\textrm{r}}} $ represents the average reflectivity, the average diffraction efficiency and the average retroreflection efficiency from 0° to 60° of incident angles in 10° step, respectively. εr and tanδ represents the permittivity and loss tangent of materials, respectively. The average retroreflection efficiency has the highest value of 46.39% when the meta-units are made of Li2Mg3CrO6 (εr = 20.25) pillars after optimization. When silicon pillars are used to form the meta-units, the corresponding value is decreased to 37.74%. Due to the relatively low permittivity of silicon (εr = 11.67), the silicon pillars are the highest to cover the phase delay from 0 to 2π when the period is fixed. Higher pillars lead to stronger coupling from neighboring meta-units, which induces an additional distortion of the phase profile and decreases the efficiency of the retroreflector. When the meta-units are Ca-MgO-TiO2 pillars, the corresponding value is 33.47%. The material with large permittivity (εr = 84.7) introduces more high order resonant peaks into the spectrum for the corresponding pillar arrays. To avoid those resonant peaks around the working frequency, we must reduce the period of meta-units to 0.4 mm. However, small meta-units with high permittivity pillars also decrease the total transmittance of the metalens, thus reduce the efficiency of the retroreflector.
Table 1. The performances of retroreflectors consisting of meta-units with different materials.
Our design makes progress compared with other works, which realize retroreflectors in a continuous range of incident angles based on metasurface. In our designed retroreflector, a quadratic metalens with an NA of 0.8 is adopted, which has the theoretical FOV of 180°. However, the maximum operating angle of the retroreflector is 60° as shown in Fig. 4(e) and Table 2, which is limited by the phase distortion caused by the oblique incidence. The retroreflector in Ref. [13] has the total diameter of 600 µm and only responses to a light beam with the diameter of 34 µm, i.e., the working aperture of (1/17) D (D represents the diameter of the retroreflector). Large aperture will induce unwanted diffraction by the reflected metasurface and reduce the retroreflection efficiency. The working aperture of the retroreflector in Ref. [14] is equal to the diameter, but its range of operating angle is narrow because the hyperbolic metalens cannot keep the focusing beam symmetrical about the chief ray for larger incident angles. For our work, only the EM waves inside the effective aperture (Based on Eq. (4), Deff = 30 mm in this work) can propagate to the focal point on the reflector. Considering the diameter of the retroreflector D = 40 mm, the working aperture is equal to (3/4) D.
Table 2. The performances of retroreflectors based on metasurface
We have discussed the retroreflection efficiency ηr normalized to the incident power within the effective aperture. To evaluate the retroreflection efficiency fairly, we re-calculate the retroreflection efficiency within the physical aperture $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ and consider different polarizations of the incident waves. Figure 6 shows ηr and $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ under two polarizations of the incident wave. The red dashed and black solid lines represent the retroreflection efficiencies ηr and $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$, respectively. Lines with the triangle and the square symbol represent transverse magnetic (TM) and transverse electric (TE) polarized incident waves, respectively. $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ is lower than ηr because, in the area out of the effective aperture, the incident waves are converted into the evanescent waves. The designed retroreflector can work in both polarizations of EM wave, but the retroreflection efficiency for TE polarization is lower than that for TM polarization at incident angles from 0° to 50°. This effect could be attributed to the different gaps between meta-units along the x- and y-direction. Along the x-direction, the gap between two different pillars is larger/smaller than it between two big/small pillars along the y-direction due to the different boundary conditions for two directions. It leads to different near field coupling efficiencies between meta-units under the incident wave with different polarizations. Figure 7 shows the normalized amplitude of the electric field for retroreflector under TE polarization incident waves at an incidence angle of θ = 0° and θ = 60°, respectively. The continuity of interference fringes is worse than that in Figs. 3(c)-(d), which is probably due to the distortion of transmitted wavefront.
Fig. 6. The retroreflection efficiency ηr and $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ of the designed retroreflector under the incident waves with TE/TM polarization.
Fig. 7. The electrical field distributions in xz-plane of the retroreflector under (a) the normal incidence and (b) the oblique incidence (θ = 60°) with TE polarization. The white lines show the position of the quadratic metalens and PEC reflector respectively.
Finally, a circular retroreflector with the same quadratic phase profile along the radial direction is demonstrated under the large incident angle (θ = 60°). Figure 8(a) shows the far-field power distribution of reflected waves when the device (diameter is 4 cm, thickness is 1.5 cm) has the incident wave with an angle of 60°. Most reflected EM waves are propagating back along the incident direction. The retroreflection efficiency $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ is equal to 11.51%. Subsequently, we double the diameter and thickness of the retroreflector (diameter is 8 cm, thickness is 3 cm), and the reflected waves can be further concentrated in the direction of the incidence as shown in Fig. 8(b). The efficiency $\mathrm{\eta }_\textrm{r}^\mathrm{^{\prime}}$ is equal to 22.53%, which is slightly lower than that of the cylindrical cases as presented previously.
Fig. 8. The far-field power distribution of reflected waves under the oblique incidence (θ = 60°) for circular retroreflectors with different size. The results in (a) and (b) are calculated using retroreflector with diameter of 4 cm and thickness of 1.5 cm, and with diameter of 8 cm and thickness of 3 cm, respectively. The red line shows the far-field power distribution at the xz-plane. θin, θre, and θn represent the direction of incidence, retroreflection, and normal reflection respectively.
4. Conclusions
In summary, we demonstrate a retroreflector at 77 GHz, which consists of a high NA quadratic metalens and a flat metallic reflector. The metalens is a high permittivity dielectric pillar array encoded with a quadratic phase profile. It is embedded in the top side of a low permittivity substrate and its focal plane is at the bottom side of the substrate where the metallic reflector is located. We realize a wide-angle and high-efficiency flat retroreflector through the parameter (P, H and ε) optimization of meta-units. When the meta-units have the period of 1 mm, the height of 2.8 mm and the permittivity of 20.25 (Li2Mg3CrO6), the corresponding cylindrical retroreflector has the operating angle ranging from -60° to 60°, the retroreflection efficiency of 38.51% at θ = ±60° and the average retroreflection efficiency of 46.39%. A circular retroreflector with the retroreflection efficiency of 22.53% at θ = ±60° is also demonstrated, which has the same radial meta-units distribution as the distribution along the x-axis of the above cylindrical retroreflector. The designed flat retroreflectors have potential applications in miniature wireless communication systems.
Funding
National Natural Science Foundation of China (61731001, 61871443).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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