Depth-dependent scaling of axial distances in light microscopy

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 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 𝑛1 and 𝑛2, and the wavelength 𝜆. We compare our theoretical results with wave-optics calculations and experimental results obtained using a novel 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 which will help microscopists to correctly re-scale the axial dimension in their imaging data when working under RIM. ### Competing Interest Statement SVL, MNFH and EBW declare no conflicts of interest. DBB is employed by Delmic BV, JPH has a financial interest in Delmic B.V