RCWA in ScatteringM

• Efficient for multilayers with bottom homogeneous layers
• Simple, easy to use: we need only 3 lines for the main part of the simulation

Multilayers with bottom homogeneous layers

The RCWA method involves the semi-discretization of Maxwell ’s equations in Fourier space to compute scattering matrix (s-matrix) of layers and combine these multiple s-matrices into a global one to quantify the expected optical responses.

Although the computation of a homogeneous layer is simple, without solving the eigen decomposition, the handling stack of homogeneous layers could be a numerical issue in 3D structures as these layers involve expensive matrix algebra in order to connect to the grating layers

We introduce two techniques: vector-based computation and bottom-up construction (Li.,JOSA,1996). The vector-based computation compactly presents matrices involved in computing homogeneous layers by the vector of four diagonal elements

The bottom-up construction can reduce the elements of global S-matrix to quantify optical responses

The benchmark was implemented with plasmonic sensor.The jupyter notebooks are in Example 2 and Example 3. The optical responses of 81 wavelengths of the plasmonic sensor composed of 4 layers were simulated in water. At the common diffraction order of 5 (or Fourier truncation=11), our RCWA was 10X faster. However, it must be noted that the comparison is relative. Ref employed Julia in a node of a workstation cluster with 24 cores, while our work uses Python within a personal laptop equipped with Intel® core i7 CPU.

Simple, easy to use

Simulation

The main part of the simulation requires only 3 lines in a loop of wavelengths. Then, the optical responses are easily calculated from global S-matrix.

for wth in range(len(wavelength_range)): 
    #Update: Kx,Ky,kz_inc,Vg,Vzr,Vzt  
    Scattering.KxKy_Component(wavelength_range[wth],e_ref,e_trn[wth])      
    # S_layer       
    S_layer=[Scattering.S_Layer(Thickness_Sim[lth],ERC_CONV[lth][wth])  for lth in range(NL)] 
    # Combine for S_global
    S_global=Scattering.S_System(S_layer) 

The advanced use of vector-based computation and bottom-up construction to simulate the needed elements of global S-matrix is found in Example 3

Geometry

e_m=[e_Al,e_SiO2,e_Si,e_Ge] 
Geometry=[]
for lth in range(NL):
    layer_lth={}      
    if lth ==0: # first layer: grating
        layer_lth['Shape']=Geo.Circle
        layer_lth['e_base']=e_m[lth]
        layer_lth['e_grt']=1 # air
        layer_lth['Critical']=[Diameter]
        layer_lth['Center']=[Lx_cell/2,Ly_cell/2]
    else: # homogeneous
        layer_lth['Shape']='Homo'
        layer_lth['e_base']=e_m[lth]   
    Geometry.append(layer_lth)  

Layer view

geo_e=np.array([Plot.Geo_viz(layer_lth) for layer_lth in Geometry])
Plot.Viz_xy(geo_e,pos_layer=[0,1,2,3])
Plot.Viz_z(geo_e,Thickness_Sim)

The nanostructure is presented in pixel and the real part of layer dielectric values.