Domain Decomposition Spectral Method Applied to Modal Method: Direct and Inverse Spectral Transforms

Abstract

We introduce a Domain Decomposition Spectral Method (DDSM) as a solution for Maxwell's equations in the frequency domain. It will be illustrated in the framework of the Aperiodic Fourier Modal Method (AFMM). This method may be applied to compute the electromagnetic field diffracted by a large-scale surface under any kind of incident excitation. In the proposed approach, a large-size surface is decomposed into square sub-cells, and a projector, linking the set of eigenvectors of the large-scale problem to those of the small-size sub-cells, is defined. This projector allows one to associate univocally the spectrum of any electromagnetic field of a problem stated on the large-size domain with its footprint on the small-scale problem eigenfunctions. This approach is suitable for parallel computing, since the spectrum of the electromagnetic field is computed on each sub-cell independently from the others. In order to demonstrate the method's ability, to simulate both near and far fields of a full three-dimensional (3D) structure, we apply it to design large area diffractive metalenses with a conventional personal computer.

Keywords: metalens; metasurfaces.

Figure 1

Sketch of a 2D grating. In the framework of the modal method, the $2D$ structure is divided into 4 layers $I_z^{\left(1\right)}$, $I_z^{\left(2\right)}$, $I_z^{\left(3\right)}$ and $I_z^{\left(4\right)}$ with respect to the propagation direction that is assumed to be $(O,z)$.

Figure 2

Sketch of the direct and inverse transform of DDSM applied to the near and far fields simulation of a photonics device.

Figure 3

Flowchart and parallelization structure of the direct and inverse transform of DDSM applied to the near and far fields simulation of a photonics device.

Figure 4

Sketch of a dipole field decomposition. The fundamental strategy of the DDSM to compute the footprint of a dipole field on an arbitrary large-area surface consists in dividing the surface into elementary sub-cells. Any component of the dipole field can then be locally simulated, on each sub-cell. Here, the surface $z=-z_s$ is subdivided into $3\times 3$ sub-cells. (a) Dipole field decomposition and recomposition. (b) Sketch of simulation of metasurface under a dipole illumination.

Figure 5

Footprints of the phase (a) and the real part (b) of an $E_x$ electric dipole on the plane $z=-20\lambda$. The numerical computation is performed on the whole domain $[-0.5D_x,0.5D_x]\times [-0.5D_y,0.5D_y]$. The operating wavelength: $\lambda =0.64 \mathsf{\mu }$m, $D_x=D_y=26.67 \lambda$, truncation orders $M_x=M_y=31$.

Figure 6

Footprints of the phase (a) and the real part (b) of an $E_x$ electric dipole in the plane $z=-20 \lambda$. The whole domain $[-0.5D_x,0.5D_x]\times [-0.5D_y,0.5D_y]$ is divided into a $3\times 3$ sub-cell ${\Omega }_{ij}$. Based on the proposed parallel strategy, the numerical computation is performed on each sub-cell ${\Omega }_{ij}$. The operating wavelength: $\lambda =0.64 \mathsf{\mu }$m, $d_x=d_y=8.89 \lambda$, truncation orders $m_x=m_y=10$.

Figure 7

Footprints of the phase (a) and the real part (b) of an $E_x$ electric dipole in the plane $z=-20\lambda$. The whole domain $[-0.5D_x,0.5D_x]\times [-0.5D_y,0.5D_y]$ is divided onto a $5\times 5$ sub-cell ${\Omega }_{ij}$. The operating wavelength: $\lambda =0.64 \mathsf{\mu }$m, $d_x=d_y=5.33 \lambda$, truncation orders $m_x=m_y=6$.

Figure 8

A forward simulation of a $100\lambda$-width FPZ using the DDSM. The computation domain is subdivided into $5\times 5$ sub-domains ${\Omega }_{ij}$. (c) presents the sketch of the diffractive Fresnel lens, (d) shows the phase distribution on the top face of the lens and (e) displays the transmitted field. Numerical parameters: $\lambda =0.532$ μm, $z_s=70\lambda$, $n=31$ zones (15 rings), $D_x=D_y=100\lambda$$h=400$ nm, refractive index of the lens material $1.99$, refractive index of the substrate $1.45$, $m_x=m_y=12$, ($M_x=M_y=62$).

Figure 9

Forward simulation of a metalens consisting of sub-wavelength dielectric nanorodes with different cross-section. Power distributions of the focusing spot in the $(x,y=0,z)$ plane, $\lambda =7 \mathsf{\mu }$m, $z_s=10\lambda$, domain $\Omega =12.8571\lambda \times 12.8571\lambda =3\times 3$ sub-cell ${\Omega }_{ij}$, $h=5.57 \mathsf{\mu }$m, materials: dielectrics Si $3.6082$ deposited on material with refractive index ($2.4626$), $m_x=m_y=17$, $M_x=M_y=52$.

Figure 10

Forward simulation of a metalens consisting of sub-wavelength dielectric nanorodes with different cross-section. Power distributions of the focusing spot in the $(x,y=0,z)$ plane, $\lambda =7 \mathsf{\mu }$m, $z_s=10\lambda$, domain $\Omega =12.8571\lambda \times 12.8571\lambda =5\times 5$ sub-cell ${\Omega }_{ij}$, $h=5.57 \mathsf{\mu }$m, materials: dielectrics Si $\nu =3.6082$ deposited on material with refractive index $2.4626$, $m_x=m_y=10$, $M_x=M_y=52$.

Figure 11

Forward simulation of a $\Omega =23.3\lambda \times 23.3\lambda$-width Pachatarnam–Berry phase-based metalens. The lens is designed to have a focus length of $f=20\lambda$. The number of sub-cells and truncation orders are set to $3\times 3$ and $m_x=m_y=12$ ($M_x=M_y=37$), respectively. (b) displays the power distribution in the $(x,y=0,z)$ plane. The phase distribution of the co-polarization $E_x^{RCP}(x,y)$, $E_y^{RCP}(x,y)$ components of the emerged RCP electric field at the top face of the metalens, obtained using the direct and the inverse transform, are presented in (c,d). A $[-\pi ,\pi ]$ coverage is clearly distinguished. Numerical parameters: $\lambda =0.532$ μm, focus length $z_s=20\lambda$, $D_x=D_y=23.30\lambda$, $h=400$ nm.

Figure 12

Forward simulation of a $\Omega =30.8\lambda \times 30.8\lambda$-width Pachatarnam–Berry phase-based metalens. The lens is designed to have a focus length of $f=20\lambda$. Power distributions of the focusing spot in the $(x,y=0,z)$ plane. Phase distribution of $E_x(x,y)$ component of the cross-polarization (RCP) on the top of the PB-based metalens. The number sub-cells and truncation orders are set to $5\times 5$ and $m_x=m_y=10$, respectively. Numerical parameters: $\lambda =0.532 \mathsf{\mu }$m, focus length $z_s=20\lambda$, h = 400 nm.