## 4 Principle of Operation  
In the topology diagram shown in Figure 2, it is assumed that both X polarized and Y polarized waves can be made incident on the metasurface. The proposed device has different responses to incident X and Y polarized waves. For incident X polarized waves, transmitted waves behave as LHCP at f\({}_{1}\) and RHCP at f\({}_{2}\), whereas for incident Y polarized waves, transmitted waves behave as RHCP at f\({}_{1}\) and LHCP at f\({}_{2}\). In order to understand the operation of a dual-band LP-to-CP converter, a plane horizontal (X polarized) wave travelling in the -Z direction is made incident on the surface of the unit cell. The incident wave can be expressed by Equation (1). Magnitudes of this incident wave can be expressed by Equation (2). \[\mathbf{E_{xi}}=\mathrm{E_{xi}}\mathbf{e_{x}} \tag{1}\]  
\[\mathrm{where,\;}\mathrm{E_{xi}}=\mathrm{E_{0}}\mathrm{e}^{\mathrm{j}\mathrm{k }\alpha} \tag{2}\]  
where \(\mathbf{e_{x}}\) is the unit cell in X direction. The transmitted wave can be expressed as the sum of two components, i.e., X polarized and Y polarized, as shown in Equation (3):  
\[\mathbf{E_{t}}=\mathrm{E_{xi}}\mathbf{e_{x}}+\mathrm{E_{yt}}\mathbf{e_{y}}= \mathrm{t_{xx}}\mathrm{e}^{\mathrm{j}\mathrm{\varphi_{xx}}}\mathrm{E_{0}} \mathrm{e}^{\mathrm{j}\mathrm{k}\alpha}\mathbf{e_{x}}+\mathrm{t_{xy}}\mathrm{e }^{\mathrm{j}\mathrm{\varphi_{xy}}}\mathrm{E_{0}}\mathrm{e}^{\mathrm{j}\mathrm{ k}\alpha}\mathbf{e_{y}} \tag{3}\]  
\[\mathrm{t_{xx}}=\frac{\mathrm{E_{xt}}}{\mathrm{E_{xi}}} \tag{4}\]  
\[\mathrm{t_{xy}}=\frac{\mathrm{E_{yt}}}{\mathrm{E_{xi}}} \tag{5}\]  
Figure 4: Phase difference between X polarized and Y polarized transmitted waves with incident X polarized wave. Figure 3: Transmission characteristics of proposed structure with incident: (**a**) X polarized wave, and (**b**) Y polarized wave. where \(t_{xx}\) and \(t_{xy}\) represent transmission coefficients for X to X and X to Y polarization conversion as shown in Equations (4) and (5), respectively. \(\varphi_{xx}\) and \(\varphi_{xy}\) are phase angles corresponding to \(t_{xx}\) and \(t_{xy}\), respectively. Since the proposed structure has an anisotropic structure, the magnitudes and phasers for X polarized and Y polarized transmitted wave components may be different. However, if for a certain frequency range these transmission coefficients become comparable and their phase angles are 90\({}^{\circ}\) apart, i.e., \(t_{xx}=t_{xy}\) and \(\varphi_{xy}=2\pi\pi\pm\pi/2\), with n being an integer, then the conditions for linear-to-circular polarization conversion will be met. In order to describe the transmission conversion performance of the proposed structure, the axial ratio for the transmitted wave is calculated as given in Equation (6) [38]:  
\[\mathrm{AR}=\left(\frac{\left|t_{xx}\right|_{2}+\left|t_{xy}\right|^{2}+\sqrt {a}}{\left|t_{xx}\right|_{2}+\left|t_{xy}\right|^{2}-\sqrt{a}}\right)^{1/2} \tag{6}\]  
where a can be calculated from Equation (7) [38]:  
\[\mathrm{a}=\left|t_{xx}\right|^{4}+\left|t_{xy}\right|^{4}+2\left|t_{xx} \right|^{2}\left|t_{xy}\right|^{2}\cos\left(2\varphi_{xy}\right) \tag{7}\]  
For an ideal LP-to-CP operation, AR should be 1 (0 dB). However, for most systems, a 3-dB value of the axial ratio is acceptable. Figures 3a and 4 show that in frequency bands from 1.16 THz to 1.634 THz and 3.935 THz to 5.29 THz, the transmission coefficient magnitudes are comparable, and the phase difference between them is \(-90^{\circ}\) or \(+270^{\circ}\) with \(\pm 15^{\circ}\) variation. Thus, the conditions for linear to circular polarization conversion is fully met at some frequencies, while for a range of frequencies, it partially fulfils the requirements (in this case the transmitted wave will be slightly elliptically polarized). Nonetheless, the performance criterion for linear to circular transmission type conversion (axial ratio within 3 dB) is maintained.