—An efﬁcient iterative Ping-Pong Computational Electromagnetics (CEM) algorithm has been developed to simulate the 2D planar reﬂector antennas and metasurfaces of arbitrary shapes. Without the storage and inverse of the impedance matrix required by the Method of Moments (MoM), the Ping-Pong algorithm updates the surface cur- rent and the scattering electric ﬁeld iteratively. In particular, the ﬁrst iteration gives the surface current for the reﬂector antennas or metasurfaces on the substrates of prefect ab- sorption. Numerical simulation has been carried out for an ellipse reﬂector antenna and a cross-shaped metasurface. It has been shown that convergence occurs within 10 iterations and good agreement between the Ping-Pong algorithm and the MoM has been achieved. Index Terms —Computational Electromagnetics (CEM), Ping-Pong algorithm, Method of Moments (MoM), planar reﬂector antennas, metasurfaces. I. I NTRODUCTION Efﬁcient Computational Electromagnetics (CEM) algo- rithms are important for design of 2D planar reﬂector antennas [1] and metasurfaces [2]. However, fast and accurate simulation is challenging when the wavelength is comparable to the dimensions of the reﬂector antennas or metasurfaces because resonance phenomenon could be involved. Even worse, for electrically large structures for applications from microwave and millimeter wave to optics [3]- [64], highly efﬁcient CEM algorithms have to be developed to save the memory and computation time. Method of Moments (MoM) has been a popular CEM algorithm for PEC structure simulation [65] because only the unknown surface current coefﬁcients are needed to be stored and solved, greatly saving the memory and computation time. However, MoM still needs to store the impedance matrix and perform the inverse of the impedance matrix to obtain the solution of the surface current. In this paper, we present an iterative Ping-Pong CEM algorithm to remove all of these limitations met in the MoM, i.e., the storage and inverse of the impedance matrix. What’s *Shaolin Liao is the corresponding author. The work is partly supported by China Postdoctoral Science Founda- tion No. 2020M672488.more, the Ping-Pong algorithm works for 2D planar reﬂector antennas and metasurfaces of arbitrary shapes. Fig. 1. Planar 2D structures under study: left) an ellipse antenna; and 2) a cross-shaped metasurface. Also shown is the Ping-Pong updating process of the surface current Jand the scattering electric ﬁeld Es. II. P ROBLEM FORMULATION The electromagnetic scattering problem is shown in Fig. 1. The electromagnetic source ﬁeld is incident upon the 2D planar PEC reﬂector antenna (the ellipse antenna here) or the metasurface (cross-shaped metasurface here) and we want to solve for the the unknown surface current. In this paper, the plane wave incident ﬁeld polarized in ^y direction is used. A. The PEC Boundary Condition According to the PEC boundary condition, the total tangential electric ﬁeld vanishes on the PEC surface, ^nh Es(r) +Ei(r)i (r) = 0; (1) whereris the observation point; is the binary mask that deﬁnes the geometry area of the reﬂector antenna; the superscripts sandidenote the scattering and incident electric ﬁeld, respectively; ﬁnally, ^nis the unit normal vector of the surface. B. Forward Update: Surface Current to Scattering Field During the forward update process, the scattering elec- tric ﬁeldEscan be obtained through the convolution of the surface current Jand the electric dyadic Green’s functionGe, Es(r) =L J =j!ZZ S02 Ge(R)J(r0)dS0;(2) whereLis the forward update operator; R=jrjjr0jis the distance from the source point r0to the observation pointr;!is the angular frequency; is the permeability; and the electric dyadic Green’s function is given as Eq (3), Ge(r) =g(r)I+1 k2rrg(r); (3) andIis the identity matrix; kis the magnitude of the wave vector k; andg(r)is the scalar Green’s function, g(r) =ejkr 4r; k=jkj=!p; withbeing the permittivity. C. The Electric Field Integral Equation (EFIE) According to the boundary condition of Eq. (1) and with the help of the scattering electric ﬁeld of Eq. (2), the EFIE is obtained as follows, ^n2 41Z 11Z 1Ge(R)J(r0)dS0+Ei(r)3 5 (r) = 0:(4) D. Backward Update: Scattering Field to Surface Current In the spectral domain, the Fourier spectrum of the surface current of Eq. (4) can be solved from the Fourier spectrum of the scattering electric ﬁeld [4], Jx(kx;ky) =2 !" k2k2 y kzEs x(kx;ky) +kxky kzEs y(kx;ky)# ; Jy(kx;ky) =2 !k2k2 x kzEs y(kx;ky) +kxky kzEs x(kx;ky) ; whose inverse Fourier transform gives the surface current in the spatial domain, J(x;y) =L1n Es ==o (5) =4! j( I+1 k2" @2 @y2@2 @x@y @2 @xy@2 @x2#)h g(x;y)~Es ==(x;y)i ; whereL1is the backward update operator and the sub- script==denotes the tangential component. III. T HEITERATIVE PING-PONG ALGORITHM With the forward update from the surface current to the scattering ﬁeld of Eq. (2) and the backward update from the scattering ﬁeld to the surface current of Eq. (5), the surface current can be updated iteratively like a Ping- Pong process, Jn+1=L1n Es ==;n Ei ==o ; (6)Es n+1=L Jn+1 ; (7) where = 1 is the complementary area of the reﬂector antenna or metasurface. Also, the initial solution of the surface current is given by, J1=L1n  Ei ==o ; (8) which corresponds to the solution for the reﬂector anten- nas or metasurfaces on substrates of perfect absorption because the scattering electric ﬁeld vanishes outside of the PEC surface. Fig. 2. The Ping-Pong algorithm. IV. T HEPING-PONG ALGORITHM Fig. 2 shows the Ping-Pong algorithm: it starts with the initial surface current of Eq. (8); then it perform ping- pong process to update the scattering ﬁeld and surface current, as shown in Fig. 1 — the forward update of the scattering ﬁeld from the surface current is done according to Eq. (7); followed by the backward update of the surface current from the scattering ﬁeld according to Eq. (6); ﬁnally, this Ping-Pong process repeats until the convergence condition is met. V. N UMERICAL VALIDATION Two typical 2D PEC structures are used to show the efﬁciency of the Ping-Pong algorithm, as shown in Fig. 1: 1) an ellipse antenna with semi-major width of =2 and semi-minor height of =4; and 2) a cross-shaped metasurface with length of 1and width of =2. Also, plane wave incidence of ^ypolarization is used. Fig. 3 shows the surface current distribution (normal- ized by the wave impedance =p =) for the ellipse antenna with 4 iterations and Fig. 4 shows the comparison Fig. 3. Surface current distribution of the ellipse antenna: left) ^x componentjJxj; and right) ^ycomponentjJyj. Fig. 4. Comparison of the ellipse antenna’s surface current ( ^ycompo- nentjJyj) between our Ping-Pong algorithm and those of the : left) along the ^xdirection; and right) along the and ^ydirection. of the surface currents between the Ping-Pong algorithm (red dots) and the MoM (blue lines), along ^xand^y directions, as shown by the red dashed lines in Fig. 1. It can be seen that good agreement is achieved between the Ping-Pong algorithm and the MoM. Similarly, Fig. 5 and Fig. 6 show the surface current distribution of the cross-shaped metasurface with 6 itera- tions and its comparison between the Ping-Pong algorithm (red dots) and the MoM (blue lines), from which it can be seen that good agreement has been achieved again. VI. C ONCLUSION An efﬁcient iterative Ping-Pong algorithm has been developed to simulate the 2D planar reﬂector antennas and metasurfaces of arbitrary shapes. Without the storage and inverse of the impedance matrix required by the MoM, the Ping-Pong algorithm iteratively updates the surface current and the scattering electric ﬁeld until they converge. In particular, the ﬁrst iteration gives the solution for the 2D planar reﬂector antennas or metasurfaces on substrates of perfect absorption. What’s more, it usually takes less than 10 iterations for the Ping-Pong algorithm to converge, dramatically increasing the simulation speed. At last, it is straightforward to apply the Ping-Pong algorithm to simulate the 2D arrays of reﬂector antennas and metasurfaces. Fig. 5. Surface current distribution of the cross-shaped metasurface: left) ^xcomponentjJxj; and right) ^ycomponentjJyj.. Fig. 6. Comparison of the cross-shaped metasurface’s surface current (^ycomponentjJyj) between our Ping-Pong algorithm and those of the : left) along the ^xdirection; and right) along the and ^ydirection.