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Abstract
A systematic study of a robust angular tolerance ultra-broadband metasurface for arbitrary rotation of linear polarization is demonstrated. The proposed method combines the spin-dependent Pancharatnam-Berry phase and the generalized Snell’s law to achieve an arbitrary angle linear polarization rotator and beam splitter. Numerical results of one terahertz example show that a 90° polarization rotator has a polarization conversion ratio of more than 90% from 1.3 to 2.3 THz in the ultra-broadband range. This method represents a significant advance in versatile, flexible design and performance compared to previously reported birefringent material wave plates, grating structures, and multi-resonance-based polarization rotators.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In biology [1], analytical chemistry [2], optical sensing, imaging and communication technology [3–6], manipulation of polarization state of light plays a very important role. Compared to the conventional techniques using bulky size optical elements [7], artificial functional materials have more advantage in many aspects [8]. Advanced functional meta-devices can independently or dependently tailor electromagnetic waves’ behavior like polarization and phase without suffer from the bulky volume and a narrow band [6,9]. In the last decade, metasurfaces, a class of surface textured with length scales much smaller than the wavelength of the incident light, have attracted much attention [10]. Many reflective type metasurface achieving linear polarization rotation, e.g., rotating linear polarization (LP) incidence into LP reflection, have been proposed and widely studied, such as leaf-shaped structure [11], square split ring resonator [12], ladder-shaped structure [13], double U-shaped patches [14], H-shaped patches [15], anisotropic high-impedance surface [16] and other type structures [17–28]. Although the aforementioned designs achieve polarization conversion, however, this type of polarization rotation originates from the multiple resonance mechanism and only feasible for the cross-polarization conversion. The operating bandwidth is often intrinsically wavelength-dependent, particularly for broadband applications. Due to their functionalities highly depending on the structure’s resonances, this type of rotator requires reoptimizing or redesign for other polarization angles, i.e., non-90 degree. Furthermore, because of the coexistence of incident and reflection beams in the same direction, it is not easy to separate both of them. This mechanism also limits its practical application.
To date, to effectively manipulate the light, the key characteristics of a metasurface need to consider the phase-shift coverage range and operating frequency range. Much effort has been invested in the past few years, from studying appropriate meta-atom structures to different physical mechanisms. In 2011 Capasso’s group first introduced the discontinuities interface using V-shaped antennas, namely phase gradient metasurface [29,30], to flexible control the wavefronts. This way opens up the avenue for developing new advanced artificial structures [9,31–34]. Another technique, the geometric phase, also known as the Pancharatnam-Berry (P-B) phase, owing to Pancharatnam and Berry’s pioneering and groundbreaking work [35,36], also received intense interest in the scientific community due to their unique phase modulation [37,38]. In contrast to the traditional bulky optical element relying on phase accumulation along the propagation direction, the geometric phase is spin-dependent, i.e., the phase profile will be the opposite when the spin of the incident wave is reversed [28]. And the phase of the scattered wave can be effectively modulated through the simply in-plane orientation of the geometry [39–42]. Hence, this approach can achieve an accurate, complete phase covering, not need to change the geometry dimensions but only rotate the orientation angle of the element. Furthermore, the identical meta-atom can simultaneously maintain high conversion efficiency and quasi-uniform amplitude response, i.e., without significant relative amplitude differences at a different position, which has a unique application in the science and engineering community compared to the spatially-variant meta-atom element [43,44]. Moreover, different from the muti-resonance mechanism, using the P-B phase design can significantly increase the operation bandwidth of the metasurface and quickly realize the wavelength-independent devices. In this vein, several metasurfaces based on the P-B phase have been proposed, achieving various functions like quadruplex polarization channels chirality-assisted P-B metasurface [45], circular polarization bi-functional metalens [46], sorting of superimposed OAMs mode [39], spin-selective metasurface [47], broadband polarization deflector [48], dynamic modulation of spin and vortex beam [49,50], and broadband, high-efficiency spin-polarized modulation [51]. However, the P-B phase can only be generated under circular polarized (CP) incidence. So this approach is not feasible for the linear polarization wave phase transformation.
This paper systematically studies the new method combining the spin-dependent P-B phase and the generalized Snell’s Law to achieve an arbitrary angle linear polarization rotator and intrinsically separate both incident and reflective channels. Figure 1 shows the schematic diagram operating on the principle of the proposed method using two orthogonal circular polarization waves synthesis of linear polarization wave with the arbitrary orientation angle. Furthermore, the detail of the theoretical analysis and formulas will be demonstrated in section 2. By integrating both the propagation and geometric phases, the wavefront of the right-handed CP (RCP) and left-handed CP (LCP) reflection waves under linear polarization incident can be flexibly independently manipulated without reversing spin states. Moreover, by desirable selection phase shift $\eta$ arbitrary angle linear polarization rotator can be achieved easily. Because the P-B phase’s feature is wavelength-independent, the proposed P-B metasurface polarization rotator can work in an ultrabroad frequency band. The numerical results show that the proposed P-B phase metasurface 90$^\circ$ rotator performs well in broadband from 1.3 to 2.3 THz. The corresponding wavelength range is from 130.4$\mu$m to 230.8$\mu$m. The fractional bandwidth(FBW) is the ratio of the frequency range (highest minus lowest frequency) divided by the center frequency. And its value can achieve as much as 55.56%. The efficiency can reach as much as 90%. Furthermore, for the normal incident application, our proposed meta-atom shows the robust angular tolerance. Compared with the traditional bulk optical component and multi-resonance-based polarization rotator, the proposed metasurface rotator with only a single layer has more performance and design simplicity advantages.
Fig. 1. Schematic diagram of the proposed method. (a) Incident and reflected linear polarized waves with different inclination angles represented by Jones vectors, respectively. (b) Achieve linear polarization rotation by synthesis of two orthogonal CP waves. The sign of the red supercell gradient phase is positive, while the sign of the blue supercell gradient phase is negative.
2. Theoretical analysis and formulas
In order to reveal the underlying physics of the proposed method synthesis of the linear polarization wave with the arbitrary inclination angle using the P-B phase, we have performed theoretical analyses based on the Jones calculus [52]. Assuming the linear polarized wave illuminates the metasurface, the corresponding reflective components can be read in the following form in Eq. (1). The $r_{xx}$, $r_{yy}$, $r_{xy}$, and $r_{yx}$ are the reflection coefficients for both linear polarization along $x$- and $y$- direction, respectively. The first subscript stands for the reflection wave’s polarization state, while the second indicates the incidence wave’s polarization state. $R_{ll}$ and $R_{rr}$ are the co-polarization reflection coefficients in the circular basis set instead of the linear basis set. At the same time, $R_{rl}$ and $R_{lr}$ are the cross-polarization components, respectively. From Eq. (1b) and Eq. (1c), the abrupt phase change, namely the co-called P-B phase, could be introduced by a meta-atom with the angle of rotation $\varphi$. Because only the co-polarization components, i.e., $R_{ll}$, $R_{rr}$, contribute to the P-B phase. Hence, preserving the co-polarization components as much as possible is essential to be more efficient in utilizing the P-B phase.
Assuming the rotation angle of the unit cell concerning the $x$-axis is $\varphi$, the corresponding rotation matrix can then be given as:
Generally speaking, the orientation of supercell distribution can be arbitrarily selected. For simplicity and without loss of generality, we assume that array of the supercell distribution along the $y$-direction. Then given the generalized Snell’s Law of the reflective type metasurface, i.e., Eq. (5), the anomalous reflection angle $\theta _{r}$ can be expressed as in Eq. (6) under normal illumination.
We now discuss the design principle of synthesizing the unique linear polarized wave using two orthogonal circular polarization waves. Given that the LP wave impinging upon the reflective type metasurface can be viewed as a combination of LCP and RCP components with the same amplitude. From the Eq. (5–7), the reflected CP components by the one supercell will keep their handedness unchanged, but along with two different directions, $\pm \theta _{r}$. As shown in Fig. 1(b), the two supercells with the same unit cell have been arranged in parallel. However, their gradient phases have opposite signs, depicted as red and blue, respectively. Hence two orthogonal CP reflection waves can exist in the same direction, e.g., LCP$^{a}$ and RCP$^{b}$ shown in Figure. Lastly, a proper spatial offset $d$ will be provided as an extra phase shift to achieve linear polarization rotation. The corresponding propagation phase difference $\eta$ is described as
Assuming the electric field of the RCP and LCP wave, two components of the incident LP, is $E^{i}_{r}$ and $E^{i}_{l}$, respectively. As shown in Fig. 1, due to the opposite sign of gradient supercells the reflection RCP and LCP wave can be represented as $E^{a}_{r}$, $E^{a}_{l}$, $E^{b}_{r}$ and $E^{b}_{l}$, respectively. The general relation between the electric fields of the input and output waves is expressed using the Jones matrix of the reflection coefficients $R\left (\pm \varphi \right )$ as [54]Once two orthogonal circular polarization reflection waves are produced, we can achieve a linear polarized wave with an arbitrary inclination angle $\zeta$. The spatial offset $d$ providing a propagation phase difference between the two CP waves determines the linear polarized wave’s unique inclination angle $\zeta$. Therefore, without loss of generality, the synthesis of the LP wave using two orthogonal circular polarization waves can be described as
Case 1:
Case 2:
In fact, due to the phase angle $\eta$ in trigonometric functions covering the entire range from 0 to 2$\pi$, the Jones vectors of the $E_{o1}$ and $E_{o2}$ can be easily represented as the two circles (red and purple) depicted as in Fig. 2(a). Moreover, the Jones vectors one-to-one map to the unit circle explicitly demonstrate the inclination angle $\zeta$ range. Hence, it is easy to conclude the result that the output wave $E_{o1}$ can cover the inclination angle from 0 to $-\pi$, while the $E_{o2}$ covers the entire inclination angle from the 0 to $\pi$. It is well known that the Poincaré sphere’s equator (green dash line in Fig. 2(b)) represents all possible linear polarized wave states. The azimuthal angle of the Stokes vectors, measured from the $S_{1}$ axis, also could be depicted on Poincaré sphere as in Fig. 2(b). However, it is noted that the azimuthal angle of the Stokes vector is two times the inclination angle $\zeta$. Therefore, for the specific application that needed the arbitrary inclination angle of the linear polarized wave, we can easily design the metasurface by choosing the desirable spatial offset $d$. In other words, the proposed method can quickly achieve arbitrary continuous rotation of linear polarization waves.
Fig. 2. Synthesized wave’s polarization states. (a) The Jones vectors represent the synthesized wave’s polarization states and Jones vectors one-to-one map to the unit circle. (b) Depiction of the corresponding polarization states on Poincaré sphere.
3. P-B metasurface design, numerical results and discussion
We design and numerical analyze the P-B phase metasurface polarization rotator based on the above design principles. The schematic of the building block is demonstrated in Fig. 3. The proposed structure can efficiently reflect the CP wave without reversing the handedness while simultaneously suppressing its orthogonal spin state wave. The unit cell comprises three strips and one circular patch, as demonstrated in Fig. 3. In this study, we perform the numerical simulation using Finite difference time-domain (FDTD) in commercial software CST Microwave Studio (CST Computer Simulation Technology GmbH, Darmstadt, Germany). The Bloch boundary condition is adopted in the $x$ and $y$ direction, while the plane wave excitation is set by linearly or circularly polarized wave, as required. And the wave propagation is along the $z$-direction.
In order to obtain the fundamental characteristics of the unit cell, we need analyze the reflection amplitude and phase of the proposed unit cell under the linear polarization incident wave. Due to the unit cell is mirror-symmetric with respect to the $yz$ plane and $xz$ plane as shown in Fig. 3, the corresponding $r_{xy}$ and $r_{yx}$ in Jones matrix satisfy $r_{xy}$ = $r_{yx}$ =0. During the optimization of the unit cell, the goal is to satisfy the following condition $|r_{xx}|=|r_{yy}|=1, \phi _{xx}-\phi _{yy}=180^\circ$ [55]. The result of the optimized unit cell structure shown in Fig. 3. The upper pattern and bottom grounded plane is 200nm thick aluminum with conductivity of 3.56 $\times$ $10^{7}$ s/m. The substrate of the unit cell is a 20$\mu$m thick lossy polyimide layer, whose relative permittivity is 3.5 and loss tangent is 0.0027 [49]. The period of the unit cell is $p=80\mu$m. The other geometric parameters shown in Fig. 3 are $l_{1}=40\mu$m, $l_{2}= 33\mu$m, $l_{3}= 33.5\mu$m, $w_{1}= 3.3\mu$m, $w_{2}= 8\mu$m, $w_{3}= 4.2\mu$m, and $r= 10\mu$m. The variable $\varphi$ is defined as the rotation angle respect to the $x$-axis. Under the linear polarization wave normally incidence, the magnitude of the co-polarization reflection coefficients and cross-polarization reflection coefficients can be depicted in Fig. 4(a). The co-polarization magnitude of the reflection coefficient $|R_{xx}|$ and $|R_{yy}|$ is higher than 0.9 within a wide bandwidth ranging from 1.3 to 2.3 THz. The corresponding wavelength range is from 130$\mu$m to 230.8$\mu$m. While the cross-polarization magnitude of the reflection coefficient $|R_{xy}|$ and $|R_{yx}|$ is much less than 0.1 from 1.3 to 2.3 THz. The phase of the co-polarization reflection coefficients is depicted in Fig. 4(b). Moreover, the difference degree between them is around 180. All of these results agree well with the theoretical analysis in Eq. (7). Then, under right circularly polarized terahertz wave normal incidence, we analyze the P-B phase mechanism using the same optimized unit cell structure.
Fig. 4. Numerical results of the unit cell. (a) The magnitude of the reflection coefficients $r_{xx}$, $r_{yy}$, $r_{xy}$ and $r_{yx}$ when linear polarization wave light impinging onto the unit cell. (b) The phase degree of the reflection coefficients $r_{xx}$, $r_{yy}$ and phase difference $\phi _{xx}-\phi _{yy}$. (c) The magnitude of the reflection coefficients $R_{lr}$, $R_{rr}$ when right circular polarization wave light impinging onto the unit cell with different rotation angles. (d) The phase of the reflection coefficients $R_{rr}$ as a function of the frequency for different rotation angles. The phase of the relfection coefficients $R_{rr}$ (e) and $R_{ll}$ (f) as a function of the metaunit rotation angle for different frequencies.
It is well known that the P-B phase metasurface could manipulate the circular polarization wave states. As demonstrated previously in Eq. (1b), the unit cell’s rotation angle $\varphi$ would introduce the unique P-B phase angle $2\varphi$. So when the unit cell rotates from 0 to $\pi$, correspondingly, the phase angle will cover the entire range from 0 to $2\pi$. The proposed meta-atom keeps the co-polarization magnitude of the reflection coefficient $|R_{rr}|$ higher than 0.8 within a wide bandwidth ranging from 1.3 to 2.3 THz, as shown in Fig. 4(c). Indicating the proposed metasurface has the ultrabroad bandwidth. Furthermore, this excellent performance does not deteriorate when rotating the unit cell concerning the $x$-axis with different angles. The numerical results in Fig. 4(d) demonstrate the related phases of co-polarization reflection coefficient $R_{rr}$ after the different angle rotations, which all agree well with the previous theoretical analysis. Due to the linearity P-B phase mechanism, Fig. 4(e) and (f) show the phase as a function of the metaunit rotation angle for different selected frequencies for the RCP and LCP co-polarization conversion, respectively. We will further demonstrate the functionality of the proposed polarization rotator based on the P-B phase, with one terahertz example numerically shown in this paper.
Previously, we have demonstrated the arbitrary angle polarization rotator can be achieved by adequately designing the offset phase change and rotation orientation. Without loss of generality, a highly flexible ultra-broadband reflective-type P-B metasurface manifesting a $90^\circ$ polarization rotator in the terahertz regime is being presented in this section. The designed metasurface reflects the impinging linearly $x$-polarized electromagnetic wave as its orthogonal $y$-polarized wave from 1.3 to 2.3 THz. Figure 5(g) shows the schematic diagram of the two supercells of the present P-B phase metasurface. The transverse dimension of each supercell is 10 unit cells. And each unit cell has the same geometry but with a different rotation angle. The first-row supercell has a positive gradient change, while the second-row supercell has a negative change. Theoretical analysis indicates the $\eta$ need equal to 180$^\circ$ for cross-polarization rotation according to Eq. (1). Hence, the second row of the supercell moves along the $x$-axis at a distance of 5 unit cells, respecting the first row. The perfectly matched layer(PML) has been adopted in the FDTD simulation. The numerical results in Fig. 5(a) indicate that one of the reflective wave channels is 13.5$^\circ$ clockwise from the $y$-axis, which agrees well with the theoretical analysis using Eq. (6). Moreover, it is evidently seen that the $y$-component of the reflection E-field is much greater than the $x$- and $z$-component of the reflection E-field in Fig. 5(b, c, d), which means the functionality of the $x$-$y$ polarization rotation. Figure 5(e) and (f) present the far-field linear scaling E-field. The corresponding reference distance is 1m. It shows that the functionality of the metasurface behaves like a beam splitter. To better investigate the performance of the polarization rotator, we calculate the polarization conversion ratio (PCR). Figure 6(a) shows the corresponding PCR of the proposed metasurface with 12 unit cells in one supercell. And Fig. 6(b) gives the PCR of the metasurface with a different unit cell number per supercell at the 1.61THz. The PCR is defined for incident $x$-polarizated wave as:
These results indicate the $90^{\circ }$ polarization rotator works in a better performance. The fractional bandwidth of the metasurface can reach $55.56\%$. However, it should also be noted that due to the generic of this method synthesis, an arbitrary inclination angle of the linear polarization can be easily achieved if the other desired phase shift is provided. For example, for designing a $45^\circ$ polarization rotator, we are only required to implement the same optimized unit cell to satisfy the criterion of $\eta$ equals $90^\circ$. In comparison, it is necessary to redesign or reoptimize the geometrics of the unit cell for previously reported metasurfaces. Because their functionalities highly depend on the structure’s resonances. Hence our proposed method is more flexible and easier to achieve the high-performance linear polarization transformation.Fig. 5. Numerical results of the 90$^\circ$ polarization rotator under the $x$-polarized wave excitation. (a) Near-field reflective wave phase fronts and (b) the magnitude of $E_{y}$ at 1.61 THz. The corresponding magnitude of the (c) $E_{x}$ and (d) $E_{z}$. The meta-atom side length p is selected as the unit of measure. (e) 3D plot and (f) polar plot of the farfield linear scaling E-field, and the corresponding reference distance is 1m. (g) The schematic of two supercells of the polarization rotator with offset phase $\pi$.
Fig. 6. The polarization conversion ratio of the proposed 90$^\circ$ polarization rotator under the $x$-polarized incident wave. (a) The metasurface with 12 unit cells per supercell works at different frequencies. (b) Metasurface with a different cell number in one supercell operates at the fixed frequency of 1.61THz.
A summary of previously reported wideband metasurface reflective polarization converters are given in Table 1. Compared with the multi-resonance mechanism polarization converters only achieving the cross-polarization conversion, the proposed design has the advantage of being simultaneously broadband and flexible for designing the arbitrary inclination angle of the polarization transformer. Furthermore, our design not only has an excellent performance in the ultra-broadband range under the normal incidence but also has a robust angular tolerance. The corresponding numerical analysis of the unit cell under the different incident angles from 0 to 40 degrees is depicted in Fig. 7.
Fig. 7. Numerical results of sweeping the incident angle. The (a) magnitude and (b) phase of the reflection coefficients $R_{ll}$ as the function of the frequencies for different incident angles.
Table 1. Summary of previously reported reflective linear polarization rotator. FBW: fractional bandwidth (PCR>90%); PCP: polarization conversion property
4. Conclusion
In this paper, a single layer reflective P-B phase metasurface was presented. The device can achieve LP-to-LP rotation at an arbitrary angle with the simple setting of the phase shift angle $\eta$ and instinct split the incident and reflected beam into two different channels. We systematically demonstrate the spin-dependent P-B phase mechanism using the Jones calculus for synthesizing linear polarization waves by combining two orthogonal circular polarization waves. And proposed practical formulas can instruct the design of arbitrary inclination angle linear polarization wave rotation. A 55.56% fractional bandwidth metasurface achieving $90^\circ$ polarization rotation was designed and numerically validated in the terahertz regime. The numerical results are in good agreement with the theoretical analysis. Due to the P-B phase mechanism being wavelength-independent, metasurface retains the significant bandwidth expansion. The proposed method has generic versatility, flexibility, and excellent performance. And hence we believe it will have many potential applications in the microwave, terahertz, and optical frequencies.
Funding
National Natural Science Foundation of China (51877015, 62001068, U1831117); Natural Science Foundation of Chongqing (cstc2021jcyj-bsh0198).
Acknowledgments
X. Wu thanks the Project supported by graduate research and innovation foundation of Chongqing, China (Grant NO. CYS17039), Project NO. 2018CDYJSY0055 supported by the Fundamental Research Funds for the Central Universities and supports from the China Scholarship Council (NO.201806050092).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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