3 March 2010 Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil, and mask
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Abstract
Partially coherent imaging is formulated using two positive semi-definite matrices that include the mask as well as the source and pupil. One matrix E is obtained by shifting the pupil function as in Hopkins transmission cross coefficient (TCC) approach while the other matrix Z is obtained by shifting the mask diffraction. Although the aerial images obtained by the matrices are identical, it is shown rank(Z) ≤ rank(E) = N, where N is the number of point sources in the illumination. Therefore, less than N FFTs are required to obtain the complete aerial image. Since the matrix Z describes the signal as partitioned into eigenfunctions orthonormal in the pupil, its eigenvalues can be used to quantify the coherence through the von Neumann entropy. The entropy shows the degree of coherence in the image, which is dependent on the source, mask and pupil but independent of aberration.
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Kenji Yamazoe, Andrew R. Neureuther, "Aerial image calculation by eigenvalues and eigenfunctions of a matrix that includes source, pupil, and mask", Proc. SPIE 7640, Optical Microlithography XXIII, 76400N (3 March 2010); doi: 10.1117/12.846666; https://doi.org/10.1117/12.846666
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