Arbitrarily shaped retro-reflector by optics surface transformation Download: 893次
A retro-reflector can reflect incoming light back to its source with minimal scattering in other directions (the reflected wave is always parallel to, but in the opposite direction of, the incoming wave; see Fig.
Fig. 1. Different types of reflection: (a) specular reflection, (b) diffuse reflection, and (c) retro-reflection.
Two classical methods, designed using geometrical optics, to achieve a retro-reflection are corner-cube reflectors (three mutually perpendicular flat mirrors)[7] and cat’s eyes reflectors (two concentric hemispheres with different radii)[8]. However, for these classical retro-reflectors, the effective working angles are limited, and they are both bulk devices (not easy to integrate practically). In recent years, new ways to achieve a retro-reflection based on an Eaton lens[9] and meta-surfaces have been introduced[10]. An Eaton lens with transmuted singularity can achieve a full-angle (360 deg) retro-reflection[9]. However, it requires inhomogeneous media (refractive index is not constant), and it is still a bulk device (not a compact planar structure). Recently, a planar meta-surface retro-reflector (two cascaded meta-surfaces) has been designed and fabricated[10]. The planar shape of the meta-surface retro-reflector (very thin and lightweight) makes it very easy to integrate with planar modulators. However, its efficiency drops quickly, as the viewing angle changes from 0 deg (nearly 78%) to 50 deg (nearly 16%).
In this study, we design an arbitrary-shaped retro-reflector with high efficiency and wide working angles using a novel theory, optical surface transformation (OST)[11,12], which is a new branch of transformation optics (TO)[1315" target="_self" style="display: inline;">–
The main conclusion in OST is that any two surfaces linked by ONMs perform as equivalent surfaces because they correspond to one common surface in the reference space from the perspective of TO[11,15]. This means that points on these two surfaces have a one-to-one corresponding relation when they are linked by ONMs. The ONM, which is an extremely highly anisotropic medium, has the following relative permittivity and permeability with the main axis along the direction[15]: where diag refers to a diagonal matrix. This expression means that the ONM’s main axis is along the direction. In fact, the ONM’s main axis can be along any other direction and may change its direction inside the device[29
We first show how to use this idea to design a half-cylindrical retro-reflector and extend the principle to an arbitrary shape later. Assume the input and output surfaces of our retro-reflector are both planes of the same size [S1 and S2 in Fig.
Fig. 2. Schematic diagram using OST to design a retro-reflector. To see the one-to-one corresponding relationship in equivalent surfaces clearly, we use the gradient color to mark each equivalent surface. (a) A half-cylindrical retro-reflector: S1 and S2 are linked by ONMs with axes along the tangential direction. The distribution of the points on S1 and S2 is reversed by 180 deg, and, hence, the output beam is the retro-reflection of the incident beam. (b) A flat planar retro-reflector: ONM with main axis (the purple arrow) along the direction links S1’ and S2’. ONMs with main axis (the purple arrow) along the direction link S1 and S2 with S1’ and S2’, respectively. The directions of purple arrows indicate the directions of the electric field projection during the one-to-one transfer procedure. S1, S2, S1’, and S2’ are all equivalent surfaces. The orientations of corresponding points on S1 and S2 (gradient colored) are reversed by 180 deg.
Another explanation of the retro-reflection phenomenon is from the perspective of the wave vector. Assume the waves impinge on S1 in Fig.
Numerical simulations based on the finite element method (FEM) verify the performance of this half-cylindrical retro-reflector (see Fig.
Fig. 3. 2D numerical simulation results for the TE polarization wave case. We plot the absolute value of the normalized electric field distribution. The incident wave is a Gaussian beam with waist radius . From (a) to (h), the incident angles change from 0 deg to 70 deg.
Next, we show how to change the geometrical shape of this half-cylindrical retro-reflector to a flat planar retro-reflector. The retro-reflection effect is due to the reverse distribution of points on input and output surfaces linked by ONMs. There are many ways to connect input and output surfaces with ONMs. We can use a combination of ONMs with main axes along the and directions to achieve the same effect [see Fig.
Fig. 4. 2D numerical simulation results for the thin planar retro-reflector as the incident angle changes from (a) 0 deg to (h) 70 deg. The height and width of the planar retro-reflector are and , respectively. The incident wave is a Gaussian beam with waist radius .
Fig. 5. 2D numerical simulation results for the TE polarization case: we plot the absolute value of the normalized electric field distribution. The white regions are filled by ONMs of various shapes, whose main axis directions are indicated by black arrows. The incident wave is a Gaussian beam with waist radius . The incident angle is 45 deg in all cases except (d). The geometric shapes of the retro-reflectors are (a) an isosceles triangle, (b) a flat triangle, (c), (d) irregular surfaces, and (e) an array of isosceles triangles. Note that the input and the output surfaces can be irregular surfaces as long as they are complementary, which ensures that the phase fronts of the input and output beams are parallel. The input and output surfaces are no longer symmetric, so we simulate 45 deg incidence in (c) and incidence in (d). (e) An array of isosceles triangular retro-reflectors in (a). The retro-reflector can be very thin, e.g., in (f).
Although the geometrical shape of the designed retro-reflector can be changed, the two surfaces S1 and S2 of the designed retro-reflector linked by the ONMs have to be complementary (i.e., S1 and S2 are centrosymmetric about their common point) to ensure that the wave front of the input and output beams are parallel, i.e., no unexpected scatter waves. If the input surface S1 and the output surface S2 of the designed retro-reflector are not centrosymmetric about their common point, the efficiency of the retro-reflector will be influenced (some unexpected scatterings appear).
The flat planar retro-reflector can be designed to be very thin in theory. The key is to keep the reverse one-to-one corresponding relation between the input surface S1 and the output surface S2. Note that the efficiency of our retro-reflector remains very high (nearly 100%), as the incident angle changes from 0 deg to 30 deg. We plot the efficiency of this planar retro-reflector as the incident angle and height change in Figs.
Fig. 6. (a) Efficiency of the planar retro-reflector as incident angle changes. (b) Efficiency of the planar retro-reflector as the height changes.
We use micro-channels composed of perfect electrical conductors (PECs) and zero index materials (ZIMs), which have been proposed to realize other ONM-based devices[3638" target="_self" style="display: inline;">–
Fig. 7. Realization of the retro-reflector. (a) Basic diagram using micro-channels. (b) 2D numerical simulation results for the TM polarization wave case with an incident angle of 40 deg. (c) Efficiency of the retro-reflector with respect to different incident angles.
Based on OST and ONM, we have proposed a much simpler way to design a retro-reflector. Any two planes linked by ONMs with suitable main axes can perform as a retro-reflector: the distribution of points on one plane is reversed by 180 deg compared with the distribution of points on the other plane from the perspective of one-to-one correspondence in OST. Many retro-reflectors in various shapes have been proposed, including a half-cylinder, a thin planar slab, a flat triangle, and irregular structures. The effective working angles of the retro-reflector can be very large: the efficiency of our retro-reflector is above 98% with incident angles from to and is still above 50% even as the incident angle increases to 80 deg. All retro-reflectors designed by our method only require homogeneous anisotropic media, i.e., ONMs (the orientations of the homogeneous ONM’s main axes may be different). We have also designed micro-channels composed of PECs and ZIMs in microwave frequency to realize the proposed retro-reflector, which may have applications in navigation, sensing, tracking, measurements, and communication systems. The proposed method to design a retro-reflector for electromagnetic waves can also be directly extended to realize a multi-physical retro-reflector that can work for both electromagnetic and acoustic waves.
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Fei Sun, Yichao Liu, Yibiao Yang, Zhihui Chen, Sailing He. Arbitrarily shaped retro-reflector by optics surface transformation[J]. Chinese Optics Letters, 2020, 18(10): 102201.