The MCMS realize the function of transmissive and reflective polarization conversion. To elaborate the mechanism of the unit cell, a two-port network and the Jones matrices for transmission along the +z axis and reflection along the -z axis can be respectively given by (Lin et al., 2013) and (Fan et al., 2021) ( E x t E y t ) = T → l ( E x i E y i ) = ( t x x t x y t y x t y y ) ( E x i E y i ) (1) ( E x r E y r ) = R → l ( E x i E y i ) = ( r x x r x y r y x r y y ) ( E x i E y i ) (2) Where E ⁱ _x , E ⁱ _y , E ^r _x , E ^r _y , E ^t _x , and E ^t _y represent the electric fields of incident, transmissive, and reflective waves with x- and y-polarization. T → l represents the Jones matrix transmitted by the linearly polarized waves along the +z axis. R → l is the Jones matrix reflected by the linearly polarized incidence along the -z axis. According to the generation condition of circularly polarized waves, the transformation matrices of reflection R → c p and transmission T → c p are calculated as follows: T → c p = ( t + + t + − t − + t − − ) = 1 2 ( t x x − t y y − j ( t x y + t y x ) t x x + t y y + j ( t x y − t y x ) t x x + t y y − j ( t x y − t y x ) t x x − t y y + j ( t x y + t y x ) ) (3) R → c p = ( r + + r + − r − + r − − ) = 1 2 ( r x x − r y y − j ( r x y + r y x ) r x x + r y y + j ( r x y − r y x ) r x x + r y y − j ( r x y − r y x ) r x x − r y y + j ( r x y + r y x ) ) (4) Where r represents reflection coefficient of circular polarization for MCMS. + and - represent RCP wave and LCP wave propagating along + z axis respectively. So r ₋₊ represents the reflection coefficient of LCP wave reflected as LCP wave, the meaning of r₊₊, r_+−, and r₋₋ will not be elaborated on too much. Moreover, t represents the transmission coefficient of circular polarization for MCMS. t ₋₋ represents the transmission coefficient of LCP wave transmitted as LCP wave. According to the principle of PB phase, the transmissive circularly polarized wave can be achieved as |t _xx | = |t _yy | = 1 and phase difference Δφ _t = 180deg. It means that the metasurface can transmit the cross-circularly polarized waves and restrain the co-circularly polarized waves when |t ₊₊ | = |t ₋₋| = 0 and |t ₊₋| = |t ₋₊ | = 1. What is more, the realization conditions of co-circularly polarized reflection are |r _xx | = |r _yy | = 1 and phase difference Δφ _t = 180deg.

MCMS is simulated by the CST STUDIO 2020 with the infinite periodic boundary and the Flouquet ports and its numerical method is Finite Integration Theory (FIT). When the linearly polarized incidence occurs along + z axis, the simulated results of transmission and reflection coefficients are shown in Figure 2. It is obvious that the simulated amplitude results of reflection coefficient |r _xx | are approximately equal to that of |r _yy | from 9.35 to 9.65 GHz and the amplitude results of transmission coefficient |t _xx | and |t _yy | are more than 0.7 at the same frequency range from Figure 2A. Moreover, the reflective phase difference of 180deg can be obtained in the frequency range of 9.2–10.6 GHz and the transmissive phase difference of 180deg is realized from 8.5 to 11.2 GHz from Figure 2C. Therefore, the co-circularly polarized reflection and cross-circularly polarized transmission are achieved from 9.35 to 9.65 GHz.

The reflection and transmission coefficients of elements “0” and “1” are illumined in Figures 3, 4 with LCP and RCP incidences. On one hand, the amplitude of refection coefficient with co-circular polarization is more than 0.5 from 9.35 to 10.65 GHz in Figure 3A. The phase difference of co-circular polarization between elements “0” and “1” is about 180deg in the same frequency range from Figure 3B. These results satisfy the reflection principle of PB phase. On the other hand, the amplitude of transmission coefficient with cross-circular polarization is more than 0.7 from 9 to 9.85 GHz and their phase difference is about 180deg between element “0” and element “1”. Consequently, the elements “0” and “1” of MCMS are an excellent choice to realize the LCP transmission and RCP reflection.

The multifunctional coding metasurface has been designed based on the convolution theorem of patterns. In the design, 1024 elements have been used. The rotation angle distributions of the 1024 elements are respectively demonstrated in Figures 3A,B with the angle changing along x- and y-axis based on elements “0” and “1”. The excited source of MCMS is a linearly polarized horn antenna with frequency band of 8–12 GHz. In order to eliminate the directly transmitted beam, the phase compensation method is used in the design for MCMS. When the horn antenna is at the position of (0, 0, z _f ), the compensational phase of the element with (x _m , y _n ) position can be generally calculated by ϕ m n = k 0 ( r m n − z f ) = k 0 ( ( x m ) 2 + ( y n ) 2 − z f ) = 2 π f 0 c ( ( x m ) 2 + ( y n ) 2 − z f ) (5) Where f ₀ is the frequency and k ₀ is the wave vector. c is the speed of light. Consequently, the rotation angle distribution of phase compensation can be determined for 1024 elements in Figure 3C, when the z _f = 240 mm is chosen. Finally, the rotation angle distribution of 32 × 32 elements for MCMS is illustrated in Figure 3D according to the convolution theorem. The array of MCMS is shown in Figure 1.

The simulated results of MCMS are illustrated in Figure 5. MCMS respectively realizes the four reflective beams in Figure 5A, and four transmissive beams in Figure 5B. It is clear that the four beams with RCP are obvious in the reflective space and their gain is much more than that in the transmissive space. On the contrary, we can see the four transmissive beams with gain of 17dBi and the chaotically reflective beam for MCMS with left circular polarization. It is necessary to note that the gain of four transmissive beams with LCP is more than that of four reflective beams for 3dB because the transmission coefficients of unit cell are more than the reflection coefficients. The position of the beam can be defined by deflection angle and azimuth angle (θ _s , φ _s ). According to the generalized Snell’s law, the unidimensional deflection angle θ _u of the beams in x- or y-axis can be defined as follows θ u i = arcsin ( λ / L e ) , i ∈ [ x , y ] (6) Where λ is the wavelength and Le is the length of subarray. The deflection angle θ _s in space is calculated by θ s = s i n − 1 ( sin 2 θ u x + sin 2 θ u y ) 0.5 (7)

Furthermore, the azimuth angle φ _s is calculated by φ s = t a n − 1 ( sin ⁡ θ u y sin ⁡ θ u x ) (8)

In the design, λ = 31.6 mm at 9.5 GHz. Le = 8 × 10 = 80 mm. So the deflection angle in x-axis can be chosen as θ _ux = 23.3deg and that in y-axis is θ _uy = 23.3deg. Moreover, the theoretical deflection angle and azimuth angle of transmissive beams for MCMS in space are 33.9, 45, 135, 225, and 315deg according to the formulas (6–8), respectively. (θ _s = 33.9deg. φ _s = 45, 135, 225, 315deg). The position of the transmissive beams are (33.9deg, 45deg), (33.9deg, 135deg), (33.9deg, 225deg), and (33.9deg, 315deg). The theoretical deflection angle and azimuth angle of reflective beams are 146.1, 45, 135, 225, and 315deg respectively. (θ _s = 146.1deg. φ _s = 45, 135, 225, 315deg). The position of the reflective beams are (146.1deg, 45deg), (146.1deg, 135deg), (−146.1deg, 225deg), and (−146.1deg, 315deg). The position of beams is verified in Figures 5C,D and the simulated results agree well with the theoretical data. Two dimensional radiation patterns in the plane of φ _s = 45 and 135deg are respectively demonstrated in Figures 5E,F. From simulated results, it is found that the excellent four beams of left circular polarization are obtained by MCMS in the transmissive space and the four beams of right circular polarization with gain of 13.8dBi are achieved in the reflective space. Meanwhile, the lower side-lobe level of -12.8dB is achieved for transmission as well as the side-lobe level of -8.1dB for reflection of MCMS. The simulated deflection angles of transmissive beams are 35 and 147deg. It is necessary to note that there are only negligible differences of 1.1 and 0.9deg between the simulation and the theory. From Figures 5G,H, we can see that the axial ratio is less than 3dB for the reflective and transmissive beams. Correspondingly, when it is the y-polarized incidence, the transmissive RCP waves and reflective LCP waves are obtained for the proposed metasurface. Consequently, the MCMS can realize the dual-circularly polarized beams and the beam focusing with transmission and reflection in the whole space.