Wsr : Electromagnetic field simulator by RCWA

Calculation principle

The calculation principle of wsr is RCWA (rigorous coupled-wave analysis), which solves the Maxwell equation (Wave Matrix) in Fourier space through the processes (1)-(6) below. If the structure consists of m layers ( i=1~m), the processes (1)-(6) are carried out in each layer.


(1)  Definition of electromagnetic vectors and lattice matrix.


(2)  Convolution matrix calculation of relative permittivity and compression of the matrix.
A square matrix of (2n+1)×(2n+1) is extracted from the center of the discrete Fourier transform matrices for the permittivity distribution ε_i(x,y) and a convolution matrix of size (n+1))²×(n+1)2)² is generated. The extraction shape can be reduced to a circle or diamond, and the size of the convolution matrix can be also greatly compressed. This is a unique feature of wsr. In wsr, n (harmonics number) is set by hm (harmonics number ratio) and the extraction shape is set by trc (truncation factor). This extraction is an approximate operation in the RCWA method and should be determined in consideration of the balance with the computational load since it sacrifices computational accuracy. If the extracted shape is square, the size of the matrices 𝑷𝑖_i, 𝑸_i , 𝜴𝑖_i², and 𝑾_i becomes 2(n+1)²×2(n+1)² and the matrix size of the wave equation becomes 4(n+1)²×4(n+1)². The computational load of memory and CPU is proportional to (2n)².


(3)  Structural matrix calculation.

(4)  Solution of eigenvalue problems.

(5)   Alignment of matrices.
The matrix is aligned by switching the order of eigenvalues λ_i² to align the diffraction orders.

(6)   Solving the wave equation.

Types of light source and boundary conditions

Since the conventional RCWA can only handle light sources with a uniform distribution and a periodic boundary condition for analysis region, the analysis targets are almost limited to the calculation of reflectance, transmittance, and diffraction efficiency. Since WSR can handle distributed light sources and select an absorbed boundary condition, the analysis target is as broad as that for FDTD.

 In the case of a distributed light source and a periodic boundary condition (PBC).


 In the case of a distributed light source and an absorbing boundary condition (ABC).

Light source position

In the conventional RCWA, the light source position was only on the top surface, but in the Wsr, it can be installed at any position in the z direction.

 When the light source position is on the top surface.


 When the light source position is in the middle plane.

Calculation example for a far field

 Inclined light propagates from the top surface to the bottom surface.


 Far field pattern in the lower side.


 Far field pattern in the upper side.

Measurement of light amount

 The amount of light inflowing and outflowing to materials and the amount of light absorbed by each material can be measured individually.


 Measurement result for each material region.

Examples of cross-section for various structures

 In the case of internal definition.


 In the case of external definition using sub.dat.
The isolated structure can be defined by the four points (x1,y1), (x2,y2), (x3,y3), and (x4,y4) described in sub.dat.


The structure defined by the piled data of four points. A periodic pattern for these structures can be defined easily.

Calculation example for lens focusing

 The lens shape is expressed by stacking the internally defined circular structure.


 The lens shape is expressed by stacking the externally defined circular structure.

Calculation of diffraction efficiency.

 Light intensity distribution for uniform intensity incidence on 8-level blazed gratings under periodic boundary conditions.


 Wavelength dependence of diffraction efficiency on the model shown above.

Displaying the calculation results

 At runtime, the calculation results are displayed in real time by Wscnt.