Yamazoe Kenji
231 Cory Hall, Department of Electrical Engineering and Computer Science, University of California, Berkeley, California 94720-1770, USA.
J Opt Soc Am A Opt Image Sci Vis. 2010 Jun 1;27(6):1311-21. doi: 10.1364/JOSAA.27.001311.
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 = <phi|E|phi> = <phi|Z|phi>, 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.
本文通过引入包含光源、物体和光瞳的矩阵E和Z，对部分相干成像的两种不同矩阵表示进行了物理比较。矩阵E是通过扩展霍普金斯传输交叉系数（TCC）方法得到的，使得光瞳函数发生了偏移，而矩阵Z是通过移动物体频谱得到的。空间像I可以写成凸二次型I = <φ|E|φ> = <φ|Z|φ>，其中|φ>是表示平面波的列向量。结果表明，秩(Z) ≤ 秩(E) = 秩(T) = N，其中T是TCC矩阵，N是给定非偏振照明下的点光源数量。因此，对于完整的空间像形成，矩阵Z所需的本征函数少于N个，而矩阵E或T总是需要N个本征函数。更重要的是，秩(Z)根据由冯·诺依曼熵确定的相干程度而变化，该熵与互强度相关。对于作为物体的理想针孔，发射空间相干光，仅一个本征函数——即光瞳函数——就足以描述相干成像。在这种情况下，无论照明如何，我们都得到秩(Z) = 1且光瞳函数为唯一的本征函数。然而，即使物体是理想针孔，秩(E) = 秩(T) = N。从这个意义上说，用矩阵Z进行空间像形成在物理上比用矩阵E更有意义。矩阵Z被分解为B†B，其中B是一个奇异矩阵，这表明矩阵B以及Z是表征部分相干成像相干程度的主算子。
