1.1. Design and theory of multifunctional polarization conversion
Figure 1a, b show the schematic diagram of the proposed chiral MM structure and its working theory as a multifunctional polarization converter along two opposite propagating directions. The MM structure consists of bi-layered metallic SRR arrays separated by a polyimide dielectric layer. The schematic of one unit cell and its structural parameters are shown in Fig. 1c, in which the front SRR is rotated 90° clockwise with respect to the back one. To realize multifunctional polarization conversion, the optimized structural parameters are as follows: the outer and inner radii of the SRRs are R = 35 µm and r = 25 µm, respectively. Gap angle relative to the center of the SRR is θ = 20°. The lattice constants of one unit cell along x and y axis are Px = Py = 100 µm. The flexible t = 20 µm thick polyimide film is treated as the dielectric spacer with dielectric constant ε = 3.5. All metallic SRRs are made of gold with conductivity σ = 4.561 × 107 S/m and thickness 200 nm.
In order to study the polarization conversion property of the designed MM structure, we adopt full three-dimensional finite-difference time-domain solver of CST Microwave Studio for numerical simulations. Periodic boundary conditions are applied in the x- and y-directions, and the perfect match layer (PML) boundary conditions are imposed at the boundaries in z-direction. To ensure the accuracy, the length scale of the mesh is set to be less than or equal to λ0 / 10 throughout the simulation domain, where λ0 is the central wavelength of the incident THz radiation. In simulation, the THz waves are normally incident on the chiral MM structure.
Theories of the proposed MM structure for dual-directional and multifunctional polarization conversion is shown in Fig. 1a, b. For forward propagation (along -z-axis), a normally incident x-polarized wave is converted to transmitted LCP and RCP waves by the MM structure at resonant frequency points 0.49 THz and 0.59 THz, respectively; while a normally incident y-polarized wave along this propagating direction is converted to a transmitted x-polarized wave in a narrow-band with bandwidth 0.15 THz and central frequency 1.28 THz. On the other hand, for backward propagation (along + z-axis), due to chirality of the MM structure, a y-polarized incident wave is converted to RCP and LCP transmitted waves at 0.49 THz and 0.59 THz, respectively; while a x-polarized incident wave is converted to y-polarized transmitted wave at 1.28 THz. Therefore, by choosing appropriate polarization state and propagating direction of the incident wave, the MM structure can selectively realize linear-to-circular or linear-to-linear polarization conversion.
In the following, we will thoroughly characterize the polarization conversion performances of the chiral MM structure.
2.2 Linear-to-circular polarization conversion
In order to understand the linear-to-circular conversion performance of the chiral MM, we first simulate amplitude transmission spectra for co- and cross-polarized components and phase change between two orthogonally polarized components in the frequency range of 0.35 ∼ 0.70 THz when incident x- and y-polarized waves respectively propagate along forward and backward directions, as shown in Fig. 2a, b, where the subscripts m and n represent n-polarized incident wave and m-polarized transmitted wave. Obviously, the transmission spectra for two co-polarized components coincide with each other, and so do two cross-polarized components, meanwhile, the phase changes along two propagation directions are equal in size but opposite in sign, indicating that the MM can realize dual-directional linear-to-circular polarization conversion. However, the handedness of the circular polarization along two opposite directions are contrary to each other because the phase difference is always defined as the phase of y-polarized component minus to that of x-polarized component of the transmitted waves. In the following, we will take the incident x-polarized wave propagating along the forward direction as an example to discuss the linear-to-circular polarization conversion performance.
Assuming electric field of the incident x-polarized wave is represented by\({\vec {E}_i}={E_{xi}}{\vec {e}_x}\). After the MM structure, the electric field of the transmitted wave can be expressed as\({\vec {E}_t}={E_{xi}}({t_{xx}}{e^{j{\varphi _{xx}}}}{\vec {e}_x}+{t_{yx}}{e^{j{\varphi _{yx}}}}{\vec {e}_y})\), where txx, tyx and ϕxx, ϕyx represent amplitudes and phases of transmission coefficients for x-to-x and x-to-y polarization conversion, respectively. It has been demonstrated that when two conditions, amplitudes txx = tyx and phase difference δ = ϕyx - ϕxx = 2nπ ± 0.5π are satisfied simultaneously, the perfect linear-to-circular polarization conversion can be achieved [37].
As shown in Fig. 2a-b, at resonant frequency points 0.49 THz and 0.59 THz, the transmission amplitudes tyx and txx are almost the same and equal to 0.54 and 0.60, respectively. In addition, the phase differences δ are about 270° and 90°, respectively. Combining these results with the above mentioned conditions for linear-to-circular polarization conversion, it is proved that the transmitted waves are nearly perfect circular polarization. As the transmission amplitudes at two frequencies are larger than 0.5, the conversion efficiencies are high, indicating that the electromagnetic coupling between two metallic layers are strong.
To verify the above results and to characterize the handedness of the transmitted waves under x-polarization incidence, Stokes parameters are introduced [38, 39]:
$$\begin{gathered} \;\;\,{S_0}=t_{{xx}}^{2}+t_{{yx}}^{2}, \hfill \\ \;\;\,{S_1}=t_{{xx}}^{2} - t_{{yx}}^{2}, \hfill \\ {S_2}=2t_{{xx}}^{{}}t_{{yx}}^{{}}\cos \delta , \hfill \\ {S_3}=2t_{{xx}}^{{}}t_{{yx}}^{{}}\sin \delta . \hfill \\ \end{gathered}$$
1
Correspondingly, the normalized ellipticity e and elliptical angle χ are determined:
$$e=\frac{{{S_3}}}{{{S_0}}},\quad \sin 2\chi =\frac{{{S_3}}}{{{S_0}}}.$$
2
where e (or χ ) can be used to characterize the ellipticity and handedness of the transmitted CP waves. e = -1 (or χ = -45°) and e = + 1 (or χ = 45°) indicate that the transmitted waves are perfectly LCP wave and RCP wave, respectively [40].
Figure 2c and 2d depict the frequency responses of the ellipticity and elliptical angle for x-polarization incidence. In Fig. 2c, e is approximately equal to -1 at frequency 0.49 THz and + 1 at 0.59 THz, implying that the transmitted waves behave as LCP at 0.49 THz and RCP at 0.59 THz, respectively. Figure 2d further shows the elliptical angles χ at above two resonant frequency points, which are − 42.3° and 42.8°, respectively, close to ±45°. e ≈ ±1 or χ ≈ ±45° further demonstrate that the proposed chiral MM structure has an excellent linear-to-circular polarization conversion performance at two resonant frequency points 0.49 THz and 0.59 THz.
For y-polarization incidence along backward direction, due to the chirality of the proposed MM structure, the transmitted waves at 0.49 THz and 0.59 THz are RCP wave and LCP wave, respectively, the handedness of which are contrary to those along forward direction.
2.3 Linear-to-linear polarization conversion
To realize linear-to-linear polarization conversion, polarization state of the incident wave is y-polarization for forward direction, and x-polarization for backward direction, which are contrary to those in linear-to-circular polarization conversion. Figure 3a shows the amplitude transmission spectra for cross- and co-polarized components when y- and x-polarized wave incident from two opposite directions. Obviously, two co-polarized components coincide with each other, so do two cross-polarized components. It means that the designed MM structure can also realize dual-directional linear-to-linear polarization conversion, and the conversion performances along two opposite directions are exactly the same. Taking the incident y-polarized wave along forward direction as an example, we discuss the performance of linear-to-linear polarization conversion.
Assuming electric fields of the incident wave and transmitted wave are \({\vec {E}_i}={E_{yi}}{\vec {e}_y}\) and\({\vec {E}_t}={E_{yj}}({t_{yy}}{e^{j{\varphi _{yy}}}}{\vec {e}_y}+{t_{xy}}{e^{j{\varphi _{xy}}}}{\vec {e}_x})\), respectively, where tyy, txy and ϕyy, ϕxy are amplitudes and phases of the transmission coefficients for y-to-y and y-to-x polarization conversion, respectively. The performance of the linear-to-linear polarization conversion is often evaluated by PCR and polarization rotation angle ψ. The PCR is defined as [39, 41]:
$$PCR=\frac{{t_{{xy}}^{2}}}{{t_{{yy}}^{2}+t_{{xy}}^{2}}}.$$
3
Introducing the similar Stokes parameters as Eq. 1, the polarization rotation angle ψ is acquired [39]:
$$tg2\psi =\frac{{{S_2}}}{{{S_1}}}$$
4
As illustrated in Fig. 3a, at resonant frequency 1.28 THz, the designed MM structure exhibits quite different transmissions for the cross- and co-polarizations. The transmission of cross-polarization is nearly up to 0.80, while that of the co-polarization is less than 0.10. The remarkable difference in transmissions between two orthogonally polarized components suggests that the MM structure can realize linear-to-linear polarization conversion, that is, the incident y-polarized wave is converted to transmitted x-polarized wave for forward direction and the incident x-polarized wave is converted to transmitted y-polarized wave for backward direction. Figure 3b shows the frequency responses of PCR and polarization rotation angle ψ. It is found that in the range of 1.22∼1.37 THz, the PCR is close to 90% and ψ reaches + 90° or -90°, revealing that an incident linearly polarized wave transmitting through the MM structure can perfectly convert to its cross-polarized direction. In addition, the linear-to-linear polarization conversion can occur at a narrow-band with bandwidth 0.15 THz and central frequency 1.28 THz.
Based on former discussions, we can conclude that the proposed chiral MM structure can not only realize dual-directional linear-to-circular polarization conversion, but also realize dual-directional linear-to-linear polarization conversion. The polarization states of the output waves depend on propagating direction and polarization state of the incident waves.
2.4 Electric field vector evolution in polarization conversion
To visually elucidate the polarization conversion process of the proposed MM structure, we simulate the evolutions of electric field in the y–z plane at polarization conversion frequency points 0.49 THz, 0.59 THz, and 1.28 THz, shown as Fig. 4. In Fig. 4a and b, when an x-polarized wave (the electric field is along –x-axis) transmits through the MM structure along forward direction, electric field vectors of the transmitted waves gradually rotate clockwise to right upper and counterclockwise to right down, respectively. Obviously, the output waves are LCP and RCP waves, as the electric fields satisfy left-handed and right-handed helix properties at frequencies 0.49 THz and 0.59 THz, respectively. In Fig. 4c, at 1.28 THz, when a y-polarized wave transmits through the MM structure along forward direction, the electric field vector of the transmitted wave is converted to + x-axis direction and linear-to-linear polarization conversion realizes. Obviously, electric field evolutions of the output waves are consistent well with elliptical angle and polarization rotation angle in Fig. 2 and Fig. 3.