Analytical solution for Gamo entropy under spatially partial coherent illumination

Abstract

We present a methodology for obtaining the analytical solution of the Gamo entropy, defined by the intensity matrix proposed by Gamo [J. Opt. Soc. Am.47, 976 (1957)JOSAAH0030-394110.1364/JOSA.47.000976]. The matrix, which consists of numerous image amplitudes at all the sampling points of the entire imaging plane, is generally infinite-dimensional. The essence of our theory is that the computational difficulties arising because of the infinite-dimensionality are avoided by introducing the inner products of two image amplitudes. The integral in continuous space plays the role of a buffer against the infinite-dimensionality. The validity of the approach is confirmed by comparing our analytical solution and Yamazoe's numerical simulations [J. Opt. Soc. Am. A28, 448 (2011)JOAOD60740-323210.1364/JOSAA.28.000448].

Theoretical analysis of controllability of interference phenomena between partially coherent fields in the intensity matrix theory

“…In this paper, all of the mathematical expressions and the conventional formalism for partially coherent imaging are utilized in the same manner as in Ref. 24. We focus on the case in which the illumination pupil has a finite size.…”

“…where the i-th eigenvalue and eigenfunction of the intensity matrix are denoted by Λ i and |Ψ i 〉, respectively. Because the intensity matrix is semi-positive Hermitian, [15][16][17][18][19][20][21][22][23][24] the set of the eigenfunctions fjÉ i iji ¼ 1; . .…”

“…(49) of Ref. 24. The variables r and rA denote the distances of each of the pinholes from the center and are defined as r = |r| and rA = |r − d|.…”

“…The detailed derivation of I 12 is provided in Appendix A of Ref. 24 and was originally reported by Hopkins. 7) To compare the WDF given in Eq.…”

“…Gamo proposed a partially coherent imaging formula in a matrix form in his intensity matrix theory. [15][16][17][18][19][20][21][22][23] In our previous report, 24) we re-formulated the intensity matrix theory and presented a calculation methodology for analytically obtaining the intensity matrix eigenvalues. In that report, we utilized a specific property of the intensity matrix: the matrix can be analyzed in a manner similar to one used to examine a density matrix in quantum mechanics.…”