Two matrix approaches for aerial image formation obtained by extending and modifying the transmission cross coefficients.

This paper physically compares two different matrix representations of partially coherent imaging with the introduction of matrices E and Z, containing the source, object, and pupil. The matrix E is obtained by extending the Hopkins transmission cross coefficient (TCC) approach such that the pupil function is shifted while the matrix Z is obtained by shifting the object spectrum. The aerial image I can be written as a convex quadratic form I = = , where |phi> is a column vector representing plane waves. It is shown that rank(Z) < or = rank(E) = rank(T) = N, where T is the TCC matrix and N is the number of the point sources for a given unpolarized illumination. Therefore, the matrix Z requires fewer than N eigenfunctions for a complete aerial image formation, while the matrix E or T always requires N eigenfunctions. More importantly, rank(Z) varies depending on the degree of coherence determined by the von Neumann entropy, which is shown to relate to the mutual intensity. For an ideal pinhole as an object, emitting spatially coherent light, only one eigenfunction--i.e., the pupil function--is enough to describe the coherent imaging. In this case, we obtain rank(Z) = 1 and the pupil function as the only eigenfunction regardless of the illumination. However, rank(E) = rank(T) = N even when the object is an ideal pinhole. In this sense, aerial image formation with the matrix Z is physically more meaningful than with the matrix E. The matrix Z is decomposed as B(dagger)B, where B is a singular matrix, suggesting that the matrix B as well as Z is a principal operator characterizing the degree of coherence of the partially coherent imaging.