Realizing Generalized Brewster Effect by Generalized Kerker Effect

A simple and universal design principle to realize the generalized Brewster effect (GBE) is proposed. Based on the generalized Kerker effect (GKE), we can realize the GBE for arbitrary polarizations, any incident angles, and any frequencies on the traditional dielectric interface by the destructive interference of various of multipoles. An adjustable metallic meta-surface has been designed to achieve this effect experimentally in the microwave band. By adjusting the suspension height of meta-surfaces above the interface, the incident angles and frequency corresponding to the GBE can be modulated easily. For the two different polarized incident waves, the GBE can be realized at the same frequency and same incident angle, which shows the validity of the GKE. This design principle opens the door for realizing tunable GBE devices that can be applied in many scenarios.

Figure 1

The equivalent dipole models and interference models which can realize the GBE of arbitrary polarization based on the GKE. The red cross-circles and double-sided arrows represent equivalent electric dipoles, and the blue symbols represent equivalent magnetic dipoles. Dipoles suspended above the interface height $h_{\mathrm{gap}}$ come from the structures we add, and dipoles on the interface come from equivalence of the reflection and refraction waves. (a) The s-polarized plane wave impinges onto medium 2 at an angle. The representations of equivalent electric dipoles which vibrate in the same direction as incident electric fields. (d) There is no reflection when a p-polarized plane wave impinges onto medium 2 at angle ${\theta }_B$. (b),(c) The incidence of the s-polarized electromagnetic wave and (e),(f) the incidence of the p-polarized electromagnetic wave. The patterns in the five insets which all have a zero point in the reflection direction represent the total radiation patterns of dipoles in one unit cell, respectively.

Figure 2

Schematic and simulated results of an S ring meta-surface. (a),(b) S ring meta-surface suspends above the interface height $h_{\mathrm{gap}}$. The relative permittivity and permeability of dielectric which is full of lower half-space are ${\epsilon }_{r1}=3.5$, ${\mu }_{r1}=1$, respectively. (c) The reflectance $R_s$ as a function of azimuthal angle of incidence $\phi$ and frequency when an s-polarized plane wave impinges at angle ${\theta }_i={62}^{\circ }$. The white dashed line indicates the GBA of s polarization from $\phi =0^{\circ }$ to $\phi ={90}^{\circ }$. (d) The reflectance $R_p$ as a function of azimuthal angle of incidence $\phi$ and frequency when a p-polarized plane wave impinges at incident angle ${\theta }_i={62}^{\circ }$. (e) At the frequency of realization of the GBE corresponding to the white dashed line as shown in Fig. 2, the reflectance of s and p polarization as a function of incident angle, respectively, when incident planes are $\phi =0^{\circ },{45}^{\circ },{90}^{\circ }$, respectively. (f) At $\phi =0^{\circ }$, $\theta ={62}^{\circ }$, $f=28.5\mathrm{GHz}$ [brown circle in Fig. 2], the electric current density distributions of an S ring when a dual-polarized GBE is realized. (g) The radiation pattern A (brown) of a multipole expansion of the volume electric current density in the first schematic excited by s polarization as shown in Fig. 2. The radiation pattern B (gray) of the electric dipole equivalent to the reflection plane wave. The total radiation pattern C (yellow) obtained by the sum of patterns A and B.

Figure 3

Variations of the s-polarized GBA through the adjustments of $h_{\mathrm{gap}}$ in the incident plane $\phi =0^{\circ }$ for the structure as shown in Fig. 2. (a) Reflectance as a function of angle of incidence $\theta$ corresponding to $h_{\mathrm{gap}}=1.1,1.3,1.8,4.7$ mm. GBAs are ${\theta }_{\mathrm{GB}}={10}^{\circ },{35}^{\circ },{60}^{\circ },{85}^{\circ }$ at relevant frequencies, respectively. (b) For the four different situations in which the GBE was realized as shown in Fig. 3, the total radiation patterns respectively come from the sum of radiated far-field of multipole expansion of the structure’s volume density and equivalent electric dipole of the reflection wave.

Figure 4

Schematic and simulated results of ${90}^{\circ }$ symmetrical S ring meta-surface. (a),(b) The S ring structures whose parameters are listed in Table 1 are arranged with ${90}^{\circ }$ rotational symmetry periodically. (c),(d) The reflectance as a function of azimuthal angle of incidence $\phi$ and frequency when the s- or p-polarized plane wave impinges at angle $\theta ={62}^{\circ }$. The white dashed line indicates the GBA of s polarization from $\phi =0^{\circ }$ to $\phi ={90}^{\circ }$. (e) At $f=28.85\mathrm{GHz}$, the reflectance of two polarization waves as functions of the incident angle $\theta$ when incident planes are $\phi =0^{\circ },{45}^{\circ },{90}^{\circ }$, respectively.

Figure 5

Schematic and simulated results of the dual-layer S ring meta-surface. (a) The S ring structures are arranged in two layers periodically. The arrangement of bottom-layer structures is the same as Fig. 2, but the top-layer structures are arranged perpendicular to the interface. The inset shows a side view of the relative position of two rings in one unit. The suspension height of rings A and B are $h_{\mathrm{gap2}}$ and $h_{\mathrm{gap}}$, respectively. (b) The volume electric current density of rings A and B when the two polarization plane waves impinge at an angle $\theta ={20}^{\circ }$ corresponding to the situation of Fig. 5. The two figures on the left show electric current density for an s-polarized wave. The two figures on the right show electric current density for a p-polarized wave. (c) The reflectance as a function of the incident angle when the GBAs are $\theta ={20}^{\circ },{35}^{\circ }$, and ${50}^{\circ }$, respectively, in the incident plane $\phi =0^{\circ }$, corresponding to three different combinations of $h_{\mathrm{gap}}$ and $h_{\mathrm{gap2}}$. (d) The total radiation patterns in one unit cell when different GBAs appeared for two polarization waves as marked in Fig. 5. The red line denotes p polarization and the blue line denotes s polarization.

Figure 6

The experimental instruments and results. (a) Schematic of the experimental instruments. (b),(c) Photos of the trapezoidal prism (Epoxy FRP, ${\epsilon }_{r2}=4.4$, $\tan \delta =0.001$) to which samples and PMI foam slabs are fixed, and detailed photos of the PCB samples. In (b), the S ring planar array (marked with ①②③) which were printed on the back of PCB slab (Rogers RT $/$ duroid 5880, ${\epsilon }_{sub}=2.2$, $h=0.127$ mm, $\tan \delta =0.0009$). (c) The dual-layer structure composed of ① and ④. (d) Fixing $h_{\mathrm{gap}}=1.8$ mm, results of experiment and simulation for s- and p-polarized incident waves in incident planes $\phi =0^{\circ },{45}^{\circ }$ correspond to planar samples ② and ③ at $f=28.025$ GHz. (e) For planar sample ②, results of experiment and simulation for s-polarized incident waves in incident plane $\phi =0^{\circ }$ for $h_{\mathrm{gap}}=1.2$ mm, $1.4$ mm, and $3$ mm. (f) When placing the samples ① and ④, results of experiment and simulation for s- and p-polarized incident waves in incident plane $\phi =0^{\circ }$ through reasonable free controls of $h_{\mathrm{gap}}$ and $h_{\mathrm{gap2}}$.

Figure 7

The main situation discussed in this work is that the upper half-space and lower half-space are full of the different dielectrics whose permittivities are ${\epsilon }_d$ and ${\epsilon }_s$, respectively. (a),(b) Refraction and reflection can be equivalent to directional radiation of a infinite ideal electric current sheet on the interface between upper and lower half-spaces (solid gray arrows). The surface electric current density on the two different infinite electric current sheets are denoted by ${\mathit{J}}_{\mathit{s}\mathit{x}}=J_{0x}\hat{x}$ and ${\mathit{J}}_{\mathit{s}\mathit{y}}=J_{0y}\hat{y}$, respectively. Here $k_d$ and $k_s$ denote wave vectors that the electromagnetic wave transmitted in the different dielectric. The equivalent processes of the s-polarized wave are shown in (a), resulting in surface electric current density along the $y$ axis direction. The equivalent processes of the p-polarized wave are shown in (b), resulting in surface electric current density along the $x$ axis direction.

Figure 8

For the case shown in Fig. 1, according to Eq. (3) from the condition of the GKE, we can obtain relations of the magnitude and phase of $m_z^{\mathrm{add}}/p_{y1}^{\mathrm{equ}}$ with respect to the incident angle and suspension height. (a) Logarithm of $|m_z^{\mathrm{add}}/p_{y1}^{\mathrm{equ}}|$ as a function of the suspension height (normalized by $k_0/\pi$) and incident angle. (b) Phase of $m_z^{\mathrm{add}}/p_{y1}^{\mathrm{equ}}$ as a function of the suspension height (normalized by $k_0/\pi$) and incident angle.

Figure 9

The design process of a pyramid-array antireflection slab (PAS) fixed on a trapezoidal prism. (a) Schematic diagram of an ideal PAS and propagation of the electromagnetic wave on the interface (incident angle is ${\theta }_i$). The period of the PAS unit cell is $p=4.7$ mm, and the relative permittivity of the material is ${\epsilon }_{r2}=4.4$. (b) For the structure system as shown in Fig. 9, the reflectance as a function of incident angle when the electromagnetic wave impinges onto vacuum at angle ${\theta }_i$. (c) Schematic diagram of the propagation process when an electromagnetic wave impinges onto a trapezoidal prism whose back fixed PAS. (d) The incident angle ${\theta }_1$ on plane A as a function of incident angle ${\theta }_{i1}$ on plane B for ${\theta }_p={70}^{\circ }$.

Figure 10

The diffraction efficiency of the structure in Fig. 4 at angle of incidence of $\theta ={62}^{\circ }$ for the two typical incident planes $\phi =0^{\circ }$ and $\phi ={45}^{\circ }$. (a),(b) The diffraction efficiency of fundamental modes and nonzero higher-order modes for the incident planes $\phi =0^{\circ }$ and $\phi ={45}^{\circ }$, respectively. (c),(d) The diffraction efficiency of all other-order reflected modes for the incident planes $\phi =0^{\circ }$ and $\phi ={45}^{\circ }$, respectively.

Figure 11

Schematic and simulated results of modified ${90}^{\circ }$ symmetrical S ring meta-surface with $p=5$ mm. (a),(b) The modified S ring structures are arranged with ${90}^{\circ }$ rotational symmetry periodically. (c),(d) The reflectance as a function of azimuthal angle of incidence and frequency when s- or p-polarized plane wave impinges at an angle $\theta ={62}^{\circ }$. The white dashed line indicates the GBA of s polarization from $\phi =0^{\circ }$ to $\phi ={90}^{\circ }$. (e),(f) At $f=28.5$ GHz, the reflectance of two kinds of polarized waves as functions of incident angle $\theta$ when incident planes are $\phi =0^{\circ }$ and $\phi ={45}^{\circ }$, respectively.