Rigorous simulation of speckle fields caused by large area rough surfaces using fast algorithms based on higher order boundary element methods
Final Report Abstract
For this project, we have developed an accelerated rigorous speckle simulator with high accuracy for three-dimensional object based on surface integral equation method or equivalently, boundary element method. Two kinds of meshing methods, lower order triangular elements and higher order quadrilateral elements, have been implemented for a direct LU- solver and triangular meshing has been implemented for the MLFMM in combination with an iterative solver of GMRES. For the direct LU-solver using higher order quadrilateral elements (8-node 10-edge), which has been developed in our previous work, we have applied an advanced algorithm to treat edge overlapped singularities via a direct integration method. With this, the computation time was reduced greatly. For the direct LU-solver using flat triangular elements, three accurate tangential formulations to combine the electric and magnetic integral equations have been implemented. The LU-solver with the two meshing methods can in principle treat three-dimensional objects (or equivalently two-dimensional surfaces) with arbitrary profiles out of complex materials, nevertheless with limited surface size. For the MLFMM solver, the three formulations without and with a preconditioner have been implemented. To find out the optimal formulation specifically for rough surfaces, two representative materials, i.e., Ag and Si, have been explored at a wavelength of 500 nm. Contrary to expectations, the optimal formulations for dielectric and metallic rough surfaces are different. For Si rough surfaces, the optical one is the non-preconditioned ICTF, while for Ag rough surfaces, it is the preconditioned formulation. We have further investigated the convergence behaviors of each formulation in terms of surface roughness and surface size (or the number of unknowns). When fitted by Nγ , the computation complexity of the MLFMM solver with respective optimal formulation is reduced from γ~3.0 of the direct LU solver to γ~1.9 for Ag rough surfaces, to γ~1.35 for Si rough surfaces, and to γ~1.3 for Si smooth surfaces, the last of which is close to the ideal value. Speckle fields from Si surfaces with a number of unknown around two million (surface size of 30 × 30 µm2) have been demonstrated at λ = 500 nm, from which the evolution of speckle fields from near field to far field in different planes in different scales with full randomization can be obtained. For dielectric rough surfaces, the calculable size of current simulator is limited by the power of the computer. For metallic rough surfaces, due to the larger negative real part of its permittivity, the preconditioned formulation still has a relative slow convergence and this could be improved by a more complex preconditioner. We have summarized the calculation parameters using the direct LU solver as well as the MLFMM solver, which can be employed as a starting point for surfaces with varied roughness and materials. We have also built up an optical measuring system for far-field BRDFs. The measured spectra agree well with our simulation. The program is open for the community (e.g. to our project partner PTB) and will be further used for our following projects.
Publications
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“3D rigorous speckle simulator using surface integral equation method and multilevel fast multiple method,” ITO Annual report for 2019/2020 (2021)
L. Fu, M. Daiber-Huppert, K. Frenner and W. Osten
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“Speckle simulator for rough surfaces using surface integral equation method and multilevel fast multiple method,” DGaO Proceedings 2021, A7, Sept. Bremen (2021)
L. Fu, M. Daiber-Huppert, K. Frenner and W. Osten
