In volume fluorescence microscopy, refractive index matching is essential to minimize aberrations. There are, however, common imaging scenarios where a refractive index mismatch (RIM) between immersion and a sample medium cannot be avoided. This RIM leads to an axial deformation in the acquired image data. Over the years, different axial scaling factors have been proposed to correct for this deformation. While some reports have suggested a depth-dependent axial deformation, so far none of the scaling theories has accounted for a depth-dependent, non-linear scaling. Here, we derive an analytical theory based on determining the leading constructive interference band in the objective lens pupil under RIM. We then use this to calculate a depth-dependent re-scaling factor as a function of the numerical aperture (NA), the refractive indices {n_1}${n_1}$ and {n_2}${n_2}$, and the wavelength \lambda$\lambda$. We compare our theoretical results with wave-optics calculations and experimental results obtained using a measurement scheme for different values of NA and RIM. As a benchmark, we recorded multiple datasets in different RIM conditions, and corrected these using our depth-dependent axial scaling theory. Finally, we present an online web applet that visualizes the depth-dependent axial re-scaling for specific optical setups. In addition, we provide software that will help microscopists to correctly re-scale the axial dimension in their imaging data when working under RIM.

中文翻译：

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光学显微镜中轴向距离的深度相关缩放

在体积荧光显微镜中，折射率匹配对于最大限度地减少像差至关重要。然而，在一些常见的成像场景中，浸没和样品介质之间的折射率不匹配 (RIM) 是不可避免的。该 RIM 导致采集的图像数据发生轴向变形。多年来，人们提出了不同的轴向缩放因子来纠正这种变形。虽然一些报告提出了与深度相关的轴向变形，但到目前为止，没有任何缩放理论能够解释与深度相关的非线性缩放。在这里，我们基于确定 RIM 下物镜光瞳中的主要相长干涉带得出了分析理论。然后，我们用它来计算与深度相关的重新缩放因子，作为数值孔径 (NA)、折射率{n_1}${n_1}$和{n_2}${n_2}$以及波长lambda$\lambda$的函数。我们将理论结果与波动光学计算以及使用不同 NA 和 RIM 值的测量方案获得的实验结果进行比较。作为基准，我们在不同 RIM 条件下记录了多个数据集，并使用我们的深度相关轴向缩放理论对这些数据集进行了校正。最后，我们提出了一个在线网络小程序，可以可视化特定光学设置的依赖于深度的轴向重新缩放。此外，我们提供的软件将帮助显微镜专家在 RIM 下工作时正确地重新调整其成像数据中的轴向尺寸。