Solving slab lamellar grating problems by the singular-value-decomposition method

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Abstract
We formulate a slab lamellar grating problem as a linear system and first use the traditional matrix-inversion technique in order to solve the system. The system can readily be solved without difficulties if its dimension is small; however, the solution may not be accurate owing to truncation. As the dimension increases, the system becomes numerically singular, which prevents solution by traditional inversion methods. We then apply the singular-value-decomposition (SVD) method in order to overcome this difficulty and thus obtain the solution. Our results demonstrate that the solution is satisfactory because it meets fundamental energy-conservation criteria and satisfies boundary conditions. Our study concludes that, for grating problems involving numerical singularities, the SVD technique is superior to the inversion technique.