  Differential phase contrast microscopy (DPC) provides high-resolution
quantitative phase distribution of thin transparent samples under multi-axis
asymmetric illuminations. Typically, illumination in DPC microscopic systems is
designed with 2-axis half-circle amplitude patterns, which, however, result in
a non-isotropic phase contrast transfer function (PTF). Efforts have been made
to achieve isotropic DPC by replacing the conventional half-circle illumination
aperture with radially asymmetric patterns with 3-axis illumination or gradient
amplitude patterns with 2-axis illumination. Nevertheless, these illumination
apertures were empirically designed based on empirical criteria related to the
shape of the PTF, leaving the underlying theoretical mechanisms unexplored.
Furthermore, the frequency responses of the PTFs under these engineered
illuminations have not been fully optimized, leading to suboptimal phase
contrast and signal-to-noise ratio (SNR) for phase reconstruction. In this
Letter, we provide a rigorous theoretical analysis about the necessary and
sufficient conditions for DPC to achieve perfectly isotropic PTF. In addition,
we derive the optimal illumination scheme to maximize the frequency response
for both low and high frequencies (from 0 to 2NAobj), and meanwhile achieve
perfectly isotropic PTF with only 2-axis intensity measurements. We present the
derivation, implementation, simulation and experimental results demonstrating
the superiority of our method over state-of-the-arts in both phase
reconstruction accuracy and noise-robustness.
