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Abstract
In conventional achromatic metasurfaces configurations where the engineering paths of phase and phase dispersion are the same, limited phase dispersion control range inevitably imposes a restriction on the broadband device size, which otherwise can be arranged in a periodic manner to achieve infinite size at a single wavelength. In addition to this, the adoption of geometric-phase structures imparts the orthogonal circular polarization conversion, whereas the birefringent propagation-phase structures bring an uncertain polarization state, thus resulting in an incomplete polarization conversion. In this study, we propose an azimuthal angle-resolved beam-deflection metasurface at the same polar angle with orthogonal phase and phase dispersion control paths. Periodic expansion is implemented to achieve unlimited device size along the phase control path and the broadband light is monotonously and azimuthally deflected to the designated polar angle with the delicate design along the dispersion path. In addition, the non-crossed polarized light arising from the incomplete geometric phase control would naturally separate from the cross-polarized light when adjusting the planar geometric parameters of the birefringent metaatoms to realize dispersion control. This design scenario may find applications in phased-array detection applications where beams are azimuthally swept to search targets at one polar angle.
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1. Introduction
Mid-wavelength infrared regime is one of the three transparent atmosphere windows in infrared which facilitates the full-time and all-weather space-to-ground applications, like communications or remote sensing [1–3]. Besides, some characteristic spectra of various materials locate within this bandwidth, such as water, carbon dioxide, sulfur oxide and nitrogen oxides [4]. Through detecting the absorption peaks, it promises the quantitative measurement of greenhouse gases and enables the accurate weather broadcast. Metasurface has demonstrated its versatility and flexibility in achieving ultra-compact and multifunctional devices, such as broadband focusing and imaging [5,6], polarization resolving [7–9], and optical computing [10–12]. Recently, achromatic metadevices have attracted much attention attributing to the fact that the inherent longitudinal dispersion in natural material have been overcome with the compensation of phase and group delay [13–15]. In chromatic dispersion engineering scenarios, the matching of group delay originates from the phase dispersion, i.e. the optical lengths for different wavelengths should be compensated to the same value. For single wavelength, the size of the array can be expanded periodically since the phase also has a period of 2π. However, the dispersion control range in achromatic design depends on the range of equivalent optical path provided by the metaatoms, which cannot be folded in period of 2π. In previous achromatic works like broadband focusing and beam deflections, larger sizes require larger group delay. Due to the limited group delay control range, the metadevices are restricted an infinite size. Some efforts have been paid to overcome the size limit, for example, through optimizing the discrete phase distributions using the inverse design algorithm [16–18]. An approximately achromatic focusing effect is achieved with large size by focusing the diffracted light onto the focal plane within the operating wavelength range but at the cost of efficiency. Another way to realize large achromatic device is to reshape the phase at different wavelengths through adding the wavelength-dependent electrical stimulations [19–21]. However, it still remains unsolved for achieving high polarization contrast and device integration.
Geometric phase structures are extensively adopted to manipulate the phase in a simple way through rotating the planar angle for single wavelength design. It is always considered to be nondispersive over broad band and have perfect handness conversion of circular polarization. However, to compensate the group delay in achromatic design, it is inevitable to further adjust the planar geometry of the Pancharatnam-Berry (PB) metaatoms to support the dispersion control. In the previous achromatic beam-deflection configurations, the structural change path coincides with the phase control path and leads to the mixing of the cross-polarized light originating from the geometric phase control and non-cross polarized light from propagation phase control [5]. This reduces the purity of output light and worsens the polarization contrast.
In this paper, we propose an azimuthal angle-resolved beam-deflection metasurface at the same polar angle with orthogonally arranged metaatoms for phase and phase dispersion control. By releasing the achromatic limitation on azimuthal angles and endowing monotonous variation versus the wavelength, we can obtain the achromatic control at the polar angle. The separate manipulations along phase and dispersion paths provides a chance to enlarge the metadevice size and improve the polarization contrast. The monotonous change along azimuth direction has the potential to be applied in aspects like wavelength resolving and targets searching.
2. Design principle
The overall design of the metasurface is shown in Fig. 1(a). The dispersion control path and the phase control path of the array are orthogonally separated to ensure the decoupling. Compared with the arrangement scenarios of conventional achromatic structures, the proposed scheme can achieve repeatable periodicity in the direction of the phase control path. Elliptical metaatoms manufactured on silicon with height of 7 µm and period of 1.7 µm are adopted provide an approximately linear phase control of the cross-polarized light with high transmittance. Metaatoms used to form the phase and phase dispersion database are simulated with finite difference time domain (FDTD) method. Plane wave with LCP is incident along positive z-direction. Periodic boundary conditions are applied along the x- and y-directions. Perfectly matched layer (PML) conditions are applied in the z directions. As to the metadevice simulation to realize achromatic polar angle focusing, we also conduct the calculation with FDTD method. Due to the orthogonal design of the phase path and the group delay path, the metadevice can be periodically extended in the x-direction. Therefore, the periodic boundary conditions are used in the x-direction while the PML conditions are applied in the y directions. The transmittance and phase change for a representative metaatom is illustrated in Fig. 1(b). The group delay coverage for the adopted geometric phase metaatom in the metasurface design is shown in Fig. 1(c). With delicate arrangement of metaatoms which both satisfy the phase and group delay demands, transmitted light with crossed polarization focuses at the same polar angle but resolves along azimuthal directions as function of wavelength.
Fig. 1. Azimuthal angle-resolved beam-deflection metasurface design. (a) Schematic of achromatically polar-angle beam-deflection device. The broadband left circularly polarized (LCP) light is incident on the metasurface. Orthogonally right circularly polarized (RCP) light transmitted is azimuthally distributed within the same polar angle as function of the frequency. X-direction of the array is designed as phase control path and y-direction group delay control path. The two paths are orthogonally manipulated to improve the polarization contrast and device size. (b) The phase and transmission spectra of one metaatom with planar size 445 nm × 615 nm. (c) The phase compensation and size parameters provided by the metaatoms adopted in the deflection metasurface, where $\mathrm{\Delta }\phi $ represents the phase difference between 85 THz and 60 THz.
In the implementation of broadband phase dispersion compensation, the accuracy of phase control cannot be maintained and thus leads to the polarization conversion efficiency decreasing. Conventional achromatic scheme introduces more geometric freedom to compensate for the dispersion caused by wavelength changes, whereas the path of geometric parameter changes coincides with the path of phase control. The non-cross polarized light is independent of the rotation angle of the metaatom in the phase path but would be focused on the dispersion path, resulting in the overlap with the cross-polarized light. In our design, the phase path and the dispersion path are separated and orthogonal to each other, which leads to the natural separation of different polarizations and thus ensures the high polarization contrast.
3. Theoretical analysis
The design principle of our metasurface is shown in Fig. 2. The deflection angle is determined by the phase gradient as implied in the generalized Snell's law [22]. For simplification, we only consider normal incidence.
Fig. 2. Operation principle of the metasurface. (a) Schematic diagram of phase gradient design in x-y plane. The red and blue lines represent isophase lines. Pink and yellow arrows represent the phase gradient directions at different wavelengths. L represents the length required for a phase change of 2π and ${\varphi _{{\lambda _i}}} $ represents the angle between the designed phase gradient and x-axis at wavelength of ${\lambda _i}$. (b) The theoretical azimuthal angle as function of the wavelength. (c) The phase distributions of the transmitted cross-polarized light for $Nu{m_y}$=13 at different frequencies, where $Nu{m_x}$ and $Nu{m_y}$ respectively represent the element numbers of the metaatoms adopted in the metasurface design. (d) The corresponding lateral phase distributions of the cross-polarized light at 3.5 µm and 4.2 µm, respectively. Each pixel in the diagram represents a metaatom in the metadevice. The purple and blue dashed lines represent approximate isophase lines of 0 and π, respectively.
Therefore, the generalized Snell's law can be written as:
where $ \theta $ is the polar angle of the deflected light, λ is the working wavelength, and $\nabla \psi ({x,y,\lambda } )$ is the phase gradient that changes with the wavelength. Here, the azimuth angle ($\varphi $) of the transmission light is included in $\nabla \psi ({x,y,\lambda } )$.We optimized the metaatoms operating in geometric and propagation phases. In order to improve the polarization contrast, the phase control path is arranged orthogonally to the dispersion control path. In the phase control path, the metaatoms can be periodically arranged to be infinite as long as the phase change 2π can be achieved. However, the phase control should be also in consistent with the phase gradient required for the achromatic design. The phase dispersion of the array is shown in Fig. 2(a). Colored arrows represent the direction of the phase gradient at different wavelengths. The phase distribution for the azimuthal angle-resolved beam-deflection and focusing can be expressed as:
From Eq. (3) we can observed that there will be drastic changes in the group delay near $\lambda = {\lambda _0}$, which needs freeform surfaces or three-dimensional configured metaatoms to fulfill the required responses. Fortunately, when λ deviates from ${\lambda _0}$, the phase response curve quickly returns to be quasi-linear. In this paper, we set ${\lambda _0}$ as 5.2 µm to ensure the linear profile of phase spectra.
The second derivative of Eq. (2) is derived to evaluate the linearity of phase distributions. It is considered as acceptable if the second derivative values approach zero. Therefore, within the operation bandwidth from 3.3 to 4.4 µm both the phase and group delay are in an acceptable level that can be fulfilled with the all-silicon metaatom database. The polar angle θ maintains the same and the azimuthal angle ${\mathrm{\varphi }_{\lambda i}}$ is determined by the phase gradient. Fig. 2(b) shows the theoretical azimuthal angle as function of wavelength at the designed polar angle. The lateral phase distribution along x direction is also illustrated in Fig. 2(c). To have a direct look at the orthogonal phase control in x-direction and dispersion control in y-direction as function of wavelength, Fig. 2(d) illustrates the transmission phases of the cross-polarized light at 3.5 µm and 4.2 µm, respectively.
4. Results and discussions
Fig. 3(a) shows the farfield intensity distributions at different wavelengths. It can be seen that the broadband incident light is deflected and focused at the same polar angle but with monotonic change along counterclockwise azimuthal direction as the wavelengths increases. The polar and azimuthal angles corresponding to the maximum values of the farfield intensities are extracted and shown in Fig. 3(b). The solid lines represent the theoretical values calculated from Eqs. (2) and (3). It can be seen that the polar angles deviate less than 1° from the designed ones and the azimuthal angles are approximately in linear from 30° to 50° in the operation bandwidth. The simulations are in well agreement with theoretical values. Since the metaatoms are made from pure silicon, the transmittance of the metasurface is around 0.5. However, the metasurface works well and the efficiency of focused beam at designed polarization state is around 90% as shown in Fig. 3(c). It’s worth mentioning that only the desired polarization is deflected to the designed angles due to the separation of phase and dispersion modulation paths.
Fig. 3. Simulation results of achromatic beam-deflection metasurface. (a) Farfield intensity distributions of the cross-polarized light at different wavelengths. (b) Comparisons between the simulated polar and azimuthal angles (dot lines) and the theoretical ones (solid lines). (c) The transmittance of the metasurface and efficiency of the cross-polarized light at the designed angles.
Jones matrix of birefringent elliptical metaatoms with mirror symmetry can be expressed as:
Convert it to the form of polarization conversion, it can be written as:
For mirror-symmetric elements, $\widetilde {{t_{xy}}} = \widetilde {{t_{yx}}}$, then:
It should be noted that from Eqs. (7) and (8) it is unrelated to θ term in the same polarization conversion. This means that even if there is a non-cross polarized light in the PB mode, it is not modulated on the phase change path which is controlled by the rotation angle of the unit. Therefore, when the phase and dispersion paths are orthogonalized, the non-cross polarized and cross-polarized light has natural separation characteristics within the operation bandwidth and leads to a high polarization contrast.
In order to have a direct evaluation of the metasurface performance on polarization isolation, Fig. 4(a) shows the farfield intensity distributions of the non-cross polarized (LCP) light, i.e. the same polarization state to the incident light. It can be seen that the LCP light are focused in y-direction which is brought in by the phase dispersion engineering. It should be noted that polar achromatic deflection is realized for cross-polarized light while co-polarized light is not purposely manipulated. For the LCP light, the transmitted phase and amplitude have not been designed. The deflection spot in the -y direction is due to the arrangement method of metaatoms. When the group delay of cross-polarized light is engineered with different metaatoms, there are phase changes and this leads to a deflection along y axis. However, it should be noted that this kind of deflection of co-polarized light does not strictly follow a uniform phase gradient for different wavelengths. We also checked the transmission spectra and found they were also non-uniform. Therefore, the deflection angle of the co-polarized light in the y-direction does not show a regular trend of decreasing with the increase of wavelength and there are also some spots in the upper half of the polar plot.
Fig. 4. High polarization conversion contrast feature of the metasurface. (a) Focused energy distribution of LCP light. It is deflected along the phase dispersion path (y direction, φ=90° or 270°) with a relatively small angle around 8°. (b) The cross-polarization conversion efficiency of the metasurface as function of the incident wavelength. (c) The polarization contrast. The spatial integral range is ±2 degrees around the maximum angles.
Due to the orthogonal arrangement of phase control metaatoms, high polarization contrast is achieved. The overall polarization conversion efficiency of incident LCP to RCP light is illustrated in Fig. 4(b). As to the quick drops at both ends of the operation bandwidth, it can be explained that in order to satisfy the requirement of polar-angle achromatism on the large group delay coverage, we adjust the planar geometries of metaatoms at the cost of polarization conversion efficiency, especially at both ends of operation bandwidth. Despite the conversion efficiency varies as function of the operation wavelength, high polarization contrast (>0.9) in the whole operation bandwidth is realized as shown in Fig. 4(c). LCP light is deflected to the negative y-direction and its deflection angle is relatively small (∼8°). This feature is often ignored in conventional achromatic metasurfaces. However, it is of importance in beam deflection applications where high polarization contrast is highly required. Compared with conventional detections in different directions through mechanical rotations, phased-array detections have advantages in fast scan without inertia, multiple searching and tracking, and anti-interference. The operation mechanism-azimuthal angle-resolved focusing performance at one single polar angle over broad bandwidth in the MWIR, allows the azimuthal beam shaping with no mechanical rotation and perfect polarization isolations. With this consideration, the metasurface design in this work may find application in phased array detections.
5. Conclusion
With this orthogonal metaatoms arrangement, we realize an azimuthal angle-resolved focusing metasurface at one single polar angle. The monotonously changing azimuthal angle ranges from 30° to 48° at polar angle 18°. The phase and phase dispersion paths are separately manipulated and the conventional restriction of the dispersion control range on the device size is partially solved. In the phase control path, the size of the metasurface can be infinitely expanded. Besides, different polarized lights are naturally separated to different direction with our broadband achromatic scheme. High polarization contrast ratio >0.9 over the whole operation bandwidth is achieved. Benefiting from the large size and excellent polarization isolation, our work may find applications in phased-array detections.
Funding
National Key Research and Development Program of China (2017YFA0205800, 2018YFA0306200); National Natural Science Foundation of China (61521005, 61874126, 61875218, 61991440, 91850208); Science and Technology Commission of Shanghai Municipality (2019SHZDZX01); Science and Technology Commission of Shanghai Municipality (18JC1420401, 20JC1416000); Shanghai Rising-Star Program (20QA1410400); Strategic Priority Research Program, Chinese Academy of Sciences (XDB43010200); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDJSSWJSC007); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2017285), Natural Science Foundation of Zhejiang Province (LR22F050004).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
The data that support the findings of this study are available on request from the corresponding author on reasonable request.
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