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20 March 2000 Modeling the light diffraction by micro-optic elements using the finite element method
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Technological advances made possible microprocessing of optical and diffractive devices of subwavelength size. Such elements may find use in holography, spectroscopy, interferometry, and optical data processing. The rigorous modeling of these devices calls for solving the basic electromagnetic Maxwell's equations. If one needs to get a far-field solution, commonly used geometrical optics approximations become inadequate, which entails the necessity of developing numerical techniques for solving Maxwell's equations. For a subwavelength characteristic size of diffractive structures, one should use a vector model for analyzing diffraction processes [1]. The vector diffraction problem may be solved analytically for some periodic structures [2]. Hence, the vector diffraction by aperiodic DOEs should be simulated using numerical techniques. These numerical techniques may then be used in working out recommendations how to improve the parameters of fabricated devices or to optimize the process of developing new ones. The majority of numerical diffraction models aimed at analyzing conducting electromagnetic scatters may be divided into differential [3], integral, and variational. With integral methods, the electromagnetic field at a space point is found as a combination of contributions to this point from source fields, taken over space or surface. Popularity of the integral methods is due to their ability to deal with unlimited field problems since the Zommerfeld radiation condition holds unconditionally in the problem statement. Furthermore, the integral methods require that only the surface field of a diffraction element be known, and not the total spatial field, thus minimizing the number of unknowns. A disadvantage of the integral methods is that they lead to fully completed matrixes and, hence, require large bulk of computer memory and great computational efforts. Note that volume integral methods are also able to simulate diffraction by nonhomogeneous DOEs. Variational methods applied to solving limit-volume tasks find the solution to Helmholtz's equation by minimizing the functional relation, as opposed to directly solving differential equations. If the finite element method is stated using the Ritz method, it is represented by a variational approximation. Its statement is simple and apply to an arbitrary homogeneous medium. However, it involves no Zommerfeld radiation condition. It is common practice to use an absorbing boundary condition which is also not free from disadvantages. There is also a hybrid method [1] which implies the application of the finite element method (FEM) to the internal DOE region where nonhomogeneities may occur, and the application of the boundary element method to a DOE-external region, with the radiation conditions to be fulfilled. The methods meet on the interface, thus satisfying the field continuity condition. This approximation represents a faithful boundary condition since values of the normal derivative of reflected field are given exactly. A disadvantage of the method lies in the nondiagonal character of the matrix system, which leads to a completely filled submatrix resulting in a greater memory resources for data storage and great computational efforts
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Dmitry V. Nesterenko and Victor V. Kotlyar "Modeling the light diffraction by micro-optic elements using the finite element method", Proc. SPIE 4002, Saratov Fall Meeting '99: Laser Physics and Spectroscopy, (20 March 2000);

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