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The Rigorous Coupled Wave Analysis (RCWA) method is commonly used to simulate diffractive optical devices, such as photonic crystal LEDs, solar cells, and gratings. Although commercial software is available, the simulation time for real devices can be many hours or even days. Therefore, it is necessary to accelerate the algorithm. Over 50% of the RCWA simulation time is taken up by repetitive solutions of large eigensystems, including eigenvalues and eigenvectors. In this paper, we present a new algorithm that entirely eliminates the need for eigensystem solutions by replacing it with a matrix exponential polynomial approximation method for medium-thickness devices. When implemented on a GPU hardware platform, the total RCWA simulation time was reduced by a factor of 7-60 compared with the classic algorithm.