In the Fresnel regime, super-resolution multi-plane phase retrieval (MPPR) often requires upsampling the captured diffraction patterns via experimental or computational means to satisfy the sampling conditions imposed by the traditional angular spectrum method which quickly becomes computationally expensive for large upsampling ratios. Furthermore, the vanilla single-beam multiple-intensity reconstruction and amplitude-phase retrieval algorithms often used in MPPR are either susceptible to noise and misalignment, or suffer from slow convergence. In this paper, we introduce a new approach to super-resolution MPPR without requiring upsampling. Super-resolution is achieved by utilizing the shifted Fresnel propagator to perform forward and backward propagation, enabling the mapping of different pixel sizes between the sample and detector plane. In addition, we incorporate the gradient descent updates of amplitude flow variants due to their accelerated convergence speed as compared to the traditional MPPR methods. Our simulation studies show that the proposed algorithms are able to reconstruct the target sample with resolution limited only by the diffraction limit despite using a camera with much larger pixel sizes, corresponding to an upsampling ratio $\ge 7.0$. We also demonstrate their robustness against noise and misalignment in experiment, outperforming the super-resolution equivalent of the standard APR algorithm in reconstruction quality and speed.

在菲涅尔机制中，超分辨率多平面相位检索 (MPPR) 通常需要通过实验或计算手段对捕获的衍射图案进行上采样，以满足传统角谱方法强加的采样条件，而传统角谱方法对于大的上采样率来说很快就会变得计算成本高昂。此外，MPPR 中经常使用的普通单波束多强度重建和幅度相位检索算法要么容易受到噪声和未对准的影响，要么收敛缓慢。在本文中，我们介绍了一种无需上采样即可实现超分辨率 MPPR 的新方法。超分辨率是通过利用移位菲涅耳传播器进行前向和后向传播来实现的，从而能够在样本和检测器平面之间映射不同的像素大小。此外，我们结合了振幅流变体的梯度下降更新，因为与传统的 MPPR 方法相比，它们的收敛速度更快。我们的模拟研究表明，尽管使用具有更大像素尺寸的相机，所提出的算法能够以仅受衍射极限限制的分辨率重建目标样本，对应于上采样率$\ge 7.0$. 我们还在实验中证明了它们对噪声和错位的鲁棒性，在重建质量和速度方面优于标准 APR 算法的超分辨率等效物。