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The magic of Fourier modal method: Why can it accurately analyze complex structures?
In modern optics and electromagnetics research, the Fourier Modal Method (FMM) has shown its unparalleled power, especially when solving scattering problems from periodic dielectric structures. For example, when studying semiconductor power devices or high-efficiency solar cells, how to use this method to obtain accurate data becomes key.
The Fourier modal method uses spatial harmonics to represent devices and fields to solve for electromagnetic patterns in complex structures.
The Fourier modal method is based on Floquet's theorem, which states that solutions to periodic differential equations can be expanded using Floquet's functions. The core of this method is to divide the complex structure into multiple uniform layers, each layer is uniform in the z direction. For curved devices with non-uniform dielectric constants, a step approximation is required. The entire problem is ultimately solved by calculating and analytically propagating electromagnetic patterns in each layer and matching boundary conditions between layers.
One of the powerful features of the Fourier modal method is the use of scattering matrix techniques to resolve boundary conditions between multi-layer interfaces.
In Fourier space, by expanding Maxwell's equations, we can transform complex partial differential equations into matrix-valued ordinary differential equations. This process greatly simplifies the difficulty of numerical calculations, especially when the frequency range to be processed is limited.
However, the Fourier modal method is not without challenges. Its application in high dielectric contrast materials may cause the Gibbs effect, which affects the accuracy of analysis. In addition, when the number of spatial harmonics is truncated, the convergence speed will be limited, so fast Fourier factorization technology (FFF) needs to be used to improve computational efficiency.
The difficulty with FFF when dealing with cross-grating devices is that the calculation requires field decomposition for all interfaces, which is not easy for devices of arbitrary shapes.
The enforcement of boundary conditions is an important challenge in Fourier modal methods. When using multiple layers, the amount of calculation required to solve simultaneously will be too large. At this time, drawing on network theory and calculating the scattering matrix becomes an effective solution. Almost all Fourier mode method scattering matrices appear to be inefficient, which requires greater caution when defining scattering parameters.
This method is widely used in the semiconductor industry, especially for the detailed analysis of periodic slit structures. Modernization of measurement technology allows the utilization of transmittance and reflectance to become more efficient and less destructive, while providing the semiconductor industry with a competitive advantage in extracting critical dimensions of structures.
By combining measured polarized reflection data with the Fourier mode method, accurate periodic structure depth and critical dimension data can be obtained.
With the help of extended wavelength range reflectometers, the Fourier mode method is indeed able to accurately measure smaller structures, especially in the wavelength range 190-1000 nm, which provides more information on the optical properties of materials and their applications. possibility. In terms of high-efficiency solar cells, the Fourier mode method has also shown its potential in improving the diffraction structure. It is combined with the OPTOS formalism for overall simulation, further improving the efficiency of solar devices.
In general, the charm of the Fourier modal method lies in its ability to analyze complex structures with high efficiency and accuracy. However, with the advancement of technology and changes in needs, whether we can continue to promote the improvement and innovation of this method in the future to adapt to more complex practical applications is a question worthy of our consideration.
