The principle for customized polarization angle deflection

For explaining linear polarization conversion transmission, let us consider an incoming y-polarized plane wave normally propagating from +z to the −z direction, the electric field of reflected wave is simply expressed as

(1)$$E^r\lpar {r\comma \;t} \rpar = \lpar R_{yy}{\rm}+ R_{xy}\rpar e^{i\lpar -kz-\omega t\rpar }$$

where ω, k, R_yy, and R_xy represent the frequency, wave vector and complex reflection coefficient, respectively. On the other hand, the transmitted wave can be divided into two parts (T_yy and T_xy) and the electric field passing through the polarization converter can be expressed as equation (2):

(2)$$E^t\lpar {r\comma \;t} \rpar = \lpar {T_{yy} + T_{xy}} \rpar e^{i\lpar kz-\omega t\rpar }$$

where T_yy and T_xy denote complex transmission coefficients of y-polarized and x-polarized transmitted waves. The amplitudes of the transmission coefficients are presented by t_yy and t_xy, respectively. The phases of the transmission coefficients are given by δ_x and δ_y. The transmitted electric field is thus to be further described in equation (3):

(3)$$E^t\lpar {r\comma \;t} \rpar = \lpar {\vec{x}t_{xy}e^{\,j\delta_x} + \vec{y}t_{yy}e^{\,j\delta_y}} \rpar e^{i\lpar kz-\omega t\rpar }$$

(4)$$\Delta = \delta _x-\delta _y$$

The phase difference Δ between the x- and y-components of the transmitted electric fields is presented by equation (4). Three possible effects are resulted under normal y-polarized incidence. (i) When Δ = ±π/2 and t_yy = t_xy, the transmitted wave belongs to circular polarization, where “+” is for left-handed circular polarization and “−” for right-handed circular polarization. (ii) When Δ = ±π/2 but t_yy ≠ t_xy, elliptical polarization transmitted wave can be obtained. (iii) When Δ = 0 or π, the transmitted field is characterized with linear polarization state. The polarization angle Φ can be worked out by

(5)$${\rm \Phi } = {\rm arctan}\displaystyle{{t_{xy}} \over {t_{yy}}}$$

The amplitude of the converted transmission coefficient is then calculated by

(6)$$t_{{\rm \Phi }y}{\rm}= \sqrt {t_{yy}^ 2 {\rm}+ t_{xy}^ 2 } $$

Design and simulated analysis

The unit cell of the proposed transmissive polarization converter is shown in Fig. 1, which is a five-layered structure. For the first layer, a metallic SRR symmetric with respect to the y-axis is adopted as the receiving antenna. Both second and fourth layers are intermediate dielectric substrates. The middle layer is a continuous metallic sheet decorated with a circular hole. The bottom layer is an equal-sized transmitting SRR, which is linked with the top one by a coaxial line through the center of the overall structure. The thickness of the dielectric substrates and the diameter of circular hole on the metal sheet mainly contribute to control the amplitude of transmission waves. On the other hand, the operating frequency bandwidth is controlled by adjusting or tailoring the dimensions of metallic SRRs. The deflected polarization angle is determined by the rotation angle of the transmitting SRR.

Fig. 1. (a) The schematic diagram of the unit cell and the front views and geometric parameters of (b) the top layer, (c) the middle layer, and (d) the bottom layer.

First, the unit cell with α = 60° is computed using commercial microwave software CST to verify 60° polarization angle deflection. Both conditions along the x- and y-directions are set as “Unit Cell,” while “Open Add Space” is adopted along the z-direction. Considering y-polarized waves normally illuminate upon the top SRRs from +z to the −z axis. After systematical optimization, the geometric parameters are finally fixed as the inner radius r ₁ = 4.5 mm, the outer radius r ₂ = 3.3 mm, the width of split l = 3.9 mm, and the width of connected line w = 1 mm, respectively. The periodicity of each unit cell is set to be P = 10 mm. Each F4B dielectric layer is as thin as 2.0 mm, the dielectric constant and loss tangent of which are ɛ_r = 2.65 and tan δ = 0.001. The diameters of circular hole and the metallic via are d_out = 3 mm and d_in = 0.65 mm.

Concluded from the results indicated in Fig. 2(a), the frequency interval for good impedance matching covers from 7.5 to 10 GHz, where the specular R_yy and R_xy in dB is lower than −10 and −30 dB, respectively. This means most energy has been transmitted through the unit cell, while only little energy has been converted into x-polarization and reflected back. Accordingly, two components T_xy and T_yy in dB are averagely −1.52 and −6.22 dB. The simulated phase distributions of T_xy and T_yy are plotted in the Fig. 2(b). Within the corresponding operation frequency band, the phase difference Δ is nearly equal to zero, which indicates the two components remain orthogonal. Thus, the T _Φy in dB can be then calculated by adding two components T_xy and T_yy. From the violet curve plotted in Fig. 2(a), it is more than −0.5 dB from 7.5 to 10 GHz. To comprehensively evaluate the conversion efficiency and eliminate dielectric loss, according to [Reference Li, Gokkavas and Ozbay17], the PCR for the proposed converter can be expressed as PCR = (t_xy ² + t_yy ²)/(t_xy ² + t_yy ² + r_xy ² + r_yy ²). The post processed PCR in the operating frequency band is more than 90%, as the red curve shown in Fig. 2(c). Additionally, the deflected polarization angle Φ can be further obtained by Φ = arctan(t_xy/t_yy). From the blue curve in Fig. 2(c), it could be seen the averaged Φ is about 59.8°, basically close to the rotation angle α = 60°. The slight errors are mainly caused by the amplitude fluctuations within the good impedance frequency range.

Fig. 2. (a) The specular reflection coefficients for R_xy, R_yy in dB and normal transmission coefficients for T_xy and T_yy in dB; (b) phase distributions for T_xy and T_yy; (c) frequency response of the T _Φy and deflected polarization angle Φ; (d), (e), and (f) specular reflection coefficients R_yy for different values of t, l, and d_out.

To investigate the influences of geometric parameters, the specular reflection coefficients R_yy for different values of t, l, and d_out are then simulated and comparatively analyzed. As indicated in Fig. 2(d)–2(f), the bandwidth of R_yy < −10 dB decrease with the increase of the dielectric thickness t from 1.0 to 3.0 mm. It is also clear to see less resonance dips with an increment in t. On the other hand, when the width of split w varies from 2.9 to 4.9 mm, the corresponding curves are gradually shift to the higher frequency regime. This mainly owes to the reductive electrical dimensions. Meanwhile, the effect of coupling between two adjacent resonances is strengthened and results in a continuous frequency band for good impedance matching. Besides, as the bridge between the receiving and transmitting SRRs, the value of the diameter d_out greatly influenced the efficiency of polarization conversion. For increasing values in d_out, the R_yy around the first resonance gradually increases more than −10 dB, whereas there is only little impact taking place at the second resonance.

It is well known that an electromagnetic field causes the excitation of an electric dipole with parallel currents and a magnetic dipole excited with anti-parallel currents. The interaction between a magnetic dipole and an electric dipole will result in strong coupling and high transmission. Thus, to further understand its broadband bandwidth, the surface currents at two resonance frequency points (f = 8.2 and 9.6 GHz) are additionally monitored and illustrated in Fig. 3. It can be obviously seen that the currents on the metallic SRRs are symmetric with respect to the y-axis and the axis oriented to the rotation angle α, respectively. For the similar current distributions, the SRRs can be regarded as the microstrip symmetric dipole antennas. In detail, at f = 8.2 GHz, the magnetic resonances are generated by anti-parallel surface currents induced along the rotated SRRs and the metallic plane, as shown in Fig. 3(c) and 3(d). However, at f = 9.6 GHz, the electric resonance is generated by parallel surface currents, as indicated in Fig. 3(g) and 3(h). As for the receiving SRRs, surface currents remain anti-parallel at both resonance frequency points, as shown in Fig. 3(a), 3(b) and 3(e), 3(f). The wide-band and high-efficiency performance results from the superposition of two polarization conversion peaks around 8.2 and 9.6 GHz corresponding to the two resonant frequencies.

Fig. 3. The surface currents on the top and bottom SRRs: (a, b) and (c, d) for f = 8.2 GHz and (e, f) and (g, h) for f = 9.6 GHz.

More intuitively, the E-field distributions at 9.6 GHz are given in Fig. 4, where it can obviously observe a deflection with respect to the polarization angle of normal incidence. For further verification, the T _Φy in dB and the corresponding polarization conversion angles from α = 10 to 90° with a 10° step are also simulated using the same simulated method. As shown in Fig. 5(a), all the T _Φy remain to be more than −0.5 dB within an identical frequency band. The deflected polarization angles are still generally equal to the corresponding rotation angles, as the colorful curves plotted in Fig. 5(b). This characteristic can provide more conveniences for meeting diverse needs in practical applications.

Fig. 4. The 3D E-field distributions at 9.6 GHz on the x–o–z plane and y–o–z plane.

Fig. 5. (a) The normal transmission coefficient T _Φy in dB and (b) the deflected polarization angle Φ distributions when the rotation angle α varies from 10 to 90°.

Experimental results and analysis

For experimental validation, three samples with α = 45, 60, 90° were fabricated by printed circuit board technology, as shown in Fig. 6. All the size is identically an area of 180 mm × 180 mm. The reflection (R_xy and R_yy) and transmission (T_xy and T_yy) coefficients were measured by a pair of horn antenna and a network analyzer (Agilent N5230C). To be out of the near field, each prototype is at least 3 m away from two horn antennas. As indicated by the reflection coefficient experimental setup in Fig. 7(a), two horn antennas are closely placed on the left side of the prototype. The in-between separation angle is lower than 6°. The transmission coefficient experimental setup is given in Fig. 7(b). It can be seen that two horn antennas are symmetrically placed on either side of the prototype. The centers of two antennas and the prototype locate at the same height. For eliminating errors from the measured environment, the results are normalized with ones without inserting prototypes. In order to achieve good polarization matching, the E-field of transmitting horn antenna in both setups is parallel to the axis of symmetry of the top SRRs. The receiving and transmitting horn antennas keep the same polarization state in the co-reflection/transmission coefficient measurement, whereas they are in the orthogonal state when measuring the cross-reflection/transmission coefficient.

Fig. 6. The front and back views for the fabricated prototypes α = 45, 60, 90°.

Fig. 7. The experimental setups for (a) reflection coefficient and (b) transmission coefficient.

Under normal y-polarized incidence, the measured T _Φy corresponding to α = 45, 60, 90° are more than −0.9, −1.3, and −0.7 dB in a frequency interval of 7.5–10 GHz, while the averaged PCRs are about 92.2, 88.9, and 91.9%, as shown in Fig. 8(a)–8(c). The deflected polarization angles Φ by post processing are averagely 43.2, 59.6, and 88.6°, basically equal to the corresponding rotation angles (45, 60, and 90°), as plotted in Fig. 8(d). The averaged PCRs in simulation are approximately 96.7, 95.1, 97.1%, while the deflected polarization angles Φ on average are 44.3, 59.3, and 89.1°. The compared differences are mainly brought in the final stack and wed. Additionally, the F4B used in fabrication is not exactly the same as that in simulation, which also causes a negative impact on the performance of the polarization conversion efficiency.

Fig. 8. (a, b, and c) The measured R_yy, T _Φy in dB and the corresponding PCR and (d) the deflected polarization angles Φ for α = 45, 60, 90°.