Phenomenological modeling of Geometric Metasurfaces

Metasurfaces, with their superior capability in manipulating the optical wavefront at the subwavelength scale and low manufacturing complexity, have shown great potential for planar photonics and novel optical devices. However, vector field simulation of metasurfaces is so far limited to periodic-structured metasurfaces containing a small number of meta-atoms in the unit cell by using full-wave numerical methods. Here, we propose a general phenomenological method to analytically model metasurfaces made up of arbitrarily distributed meta-atoms based on the assumption that the meta-atoms possess localized resonances with Lorentz-Drude forms, whose exact form can be retrieved from the full wave simulation of a single element. Applied to phase modulated geometric metasurfaces, our analytical results show good agreement with full-wave numerical simulations. The proposed theory provides an efficient method to model and design optical devices based on metasurfaces.


I. Introduction
Metasurface is a thin layer of structured surface consisting of an array of planar metallic antennas with thickness far less than the wavelength of light. By tailoring the orientation and anisotropic resonances of meta-atoms, metasurfaces can be designed to control the phase, amplitude and polarization of light at the nanoscale, which has opened door for various planar photonic and optical devices with low manufacturing complexity [1][2][3]. They have been employed for ultra-thin wave plates [4][5][6], phase gradient plates [7][8][9], planar lenses [10,11], optical vortex plates [7,12] and holograms [13][14][15]. Among various types of metasurfaces, a special type of metasurfaces that manipulate the phase of light by the orientation of the antennas have shown great potential for practical applications due to the simplicity and robustness of the phase control. The underlying mechanism of the phase control is a Pancharatnam-Berry phase in the polarization state, which leads to strong spin-orbital interaction of light [16][17][18][19]. This concept has been used to realize spin controlled directional coupler for surface waves [20,21] and broadband holograms [22,23].
Up to date, theoretical modeling of metasurfaces is mainly limited to full-wave numerical simulations based on the Finite-Difference Time-Domain (FDTD) method and the Finite Element Method (FEM). Reported analytical modeling for metasurfaces, such as surface admittance method [6,[24][25][26] and homogenization with a Lorentz-formed polarizability [23], have only been applied to uniform metasurfaces with unit cell consisting of a single plasmonic element. So far, there has been no simple analytical modeling on the diffractive efficiencies of metasurfaces with complex phase profiles. In particular, for metasurfaces used as metalens, holograms and vortex plates that consist of a large number of different meta-atoms, there have been no efficient theoretical methods that could be used to accurately predict their optical performance. In this paper, we propose a general analytical theory to model geometric metasurfaces with arbitrarily distributed phase profiles. The method is shown to agree very well with, but much more efficient than the full-wave numerical simulations.

II. Analytical Theory of Metasurface
Without loss of generality, we consider a metasurface sandwiched between the 3 homogenous and isotropic cover and substrate, as depicted in Fig. 1 denote tangential component of electric fields propagating along z  direction at the interface between the cover (substrate) and metasurface. Because the metasurface has a negligible thickness h ( h   ), the overall tangential component of the electric fields is assumed to be continuous across the metasurface. Therefore the electric fields on the interfaces of the cover and substrate satisfy For the sake of simplicity, we only consider metasurfaces made up of achiral meta-atoms and neglect the surface current induced by the magnetic fields of light. The tangential components of magnetic fields at the interfaces of the cover and substrate thus satisfy Where, χ is the electric susceptibility tensor of the metasurface. Due to their deep subwavelength sizes, each antenna can be considered as an electric dipole located at   , mn xy.
The susceptibility tensor can be written as, In Eq. (3), mn α is the local polarizability tensor of the meta-atom located at    We can express the electric field and susceptibility tensor as the sum of various spatial frequency components through a Fourier expansion x y x m y n x y x y x m y n x y x y where S is the area of the metasurface. Combining Eq. (1-3) and Eq. (6), the Fourier expansions of electric and magnetic fields at the interfaces of the cover and substrate satisfy , , , , ; x y x y x y x y , , , and can be expressed as Where, S  , S  and S n are the permittivity, magnetic permeability and refractive index of the substrate. The expression for the admittance tensor of the cover is given by replacing 'S' by 'C' in Eq. 9.
It has been shown that the diffraction efficiency of metasurface can be dramatically enhanced through a three layer design, where the plasmonic antenna layer is on top of a homogeneous dielectric substrate layer of finite thickness d and a thick ground metal layer such that the metausrface operate in reflection mode (Fig. 2b). It has been shown that, with suitably designed thickness of the substrate layer, the interplay between the antenna resonance and the Fabry-Perot resonance can lead to very high efficiency over a broad bandwidth [23].
We next extend our analytical analysis to such a multilayer design. Based on Eq. (7) and (8), the electric and magnetic fields at the interfaces of the cover and metasurface satisfy the following set of equations, , , , x y x y x y , , where S η is an effective admittance that take into account the reflection by the metal ground plane and is expressed as, ,, ln( ) 2 x y Eq. 13 shows that the effective admittance of the combined dielectric layer and the ground plane is dispersive, which helps to cancel the intrinsic dispersion of the metal dipoles. Thus, with a proper thickness of the dielectric layer, a broadband and high-efficient optical device could be realized.

III. Retrieval of Lorentzian parameters
For an antenna of a specific geometry, the Lorentzian parameters can be retrieved by fitting the full wave simulation of a simple periodic array of the antennas with our analytical theory. Here CST microwave studio software is used to for the simulation. We consider a metasurface consisting of a periodic array of identical gold nanorods with the same orientation in a square lattice with period  , as shown in Fig. 3 (a). The semi-infinite cover and substrate are air and   (15) Eq. (15) shows that only cross-circular polarizations in transmitted light can gain different additional phases for the normally incident beam with circular polarizations. The additional phases are non-dispersive geometric phases determined by the orientation angle of the meta-atoms. Fig. 3 (b-e) shows the transmission amplitude and phase of the X (Y) -polarized light normally incident from cover onto the metasurface that is schematically shown in Fig. 3 (a).
We fit the curves based on Eq. (5) and (14). The retrieved parameters of the nanorod are  5) and (14).

IV. Verification of the Analytical Theory with full-wave numerical simulations
By feeding the retrieved parameters obtained from Section III into the analytical model, we can analytically model metasurfaces consisting of antennas of arbitrary orientations for manipulating the wavefront of incident beam. The analytical modeling results are compared to full wave simulations for several representative metasurfaces.
A. Phase gradient metasurface 9 We first consider a metasurface with one-dimensional geometric-phase gradient. In a periodic unit cell, the phase gradient metasurface comprises N identical meta-atoms, whose orientations angle φ exhibits a constant gradient N  along one direction. Fig. 4 (a) shows the simulated phase gradient metasurface with N equal to 8, made up of the same Au nanorods as that in Fig.3 (a). The periods of the array along X axis and Y axis are 8 and  , respectively.
From Eq. (4) and (6) where 0  is the orientations angle of the first nanorod chosen in the periodic unit cell.
In Eq. (16), we only consider the -1, 0 and +1 order diffractive light for the phase gradient metasurface, whose wave vectors   where subscripts -1, 0 and +1 denote diffraction orders of the transmitted and reflected electric fields from the metasurface, respectively, and in  in  1  0  1  0   22  ,  ,  ,  ; , 0 , 0, 0 10 For a right circularly polarized light normally incident on the metasurface shown on Fig.4 (a) from the cover, using the retrieved parameters of the nanorod, we can obtain the reflectivity and transmittivity of the zero-order and first-order diffractive light from Eq.17 (a-c). Fig 4 (b,c) show that the analytical results agree well with those obtained from the CST simulation. Furthermore, to check the applicability of Eq. 17(a -c) for the obliquely incident light, we simulate a RCP with an incident angle of 30 o from air onto the metasurface as shown in Fig.4 (a). Figure 5 shows that the analytical results still agree well with those calculated from CST simulation for the obliquely incident beam.  Fig. 4(a).

B. Reflective metasurface
The configuration of the reflective metasurface is shown in Fig. 6  in Eq. 14. Fig. 6 (b) shows the analytical and simulated reflectivity spectra of co-polarization and cross polarization for normally incident RCP onto the three layer metasurface. It is obvious that the metasurface can work as a reflective half-wave plate with the wavelength from 1100nm to 1400nm. analytical and simulated reflectivity spectra of co-polarization and cross polarization for a normally incident RCP beam.
Next, we consider a reflective metasurface (shown in Fig.7 (a)) with a linear phase gradient.
By replacing S η with S η in Eq. 17, the analytical expressions of zero-order and first-order reflective electric fields from the metasurface can be obtained. Fig. 7 (b) shows the zero-order and first-order reflectivity obtained from both the analytical theory and CST simulation for a right circularly polarized light normally incident onto the phase-gradient metasurface. Again, the analytical results agree well with the simulated results.

V. Conclusion
In summary, we have derived a generally analytical theory of metasurfaces based on the assumptions that the tangential electric fields on both sides of the metasurface are continuous and the meta-atoms possess localized resonances with Lorentz-Drude forms. By retrieving the 14 six parameters used to describe the anisotropic polarizability tensor of a meta-atom, optical performance of geometric metasurfaces constituted by the same meta-atoms but different orientations can be accurately calculated by the analytical formulae. As the analytical results show excellent agreement with full-wave numerical simulations, the analytical theory provides an efficient method to design and model optical devices based on metasurfaces of complex phase profiles.