A topology optimization method can be used to find out the optical waveguide structures which have the desired transmission characteristics. Using the function expansion method, we can avoid the problem of a gray area, which means that some areas having intermediate refractive index between those of usable materials appear in a design region. However, so far, topology optimization has mainly been studied for structures consisting of two isotropic materials. In this paper, we study the applicability of topology optimization to structures which include three or more materials, and demonstrate the optimal design of a waveguide crossing.
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