Degradation of multiphoton signal and resolution when focusing through a planar interface with index mismatch: Analytical approximation and numerical investigation

Multiphoton microscopy (MPM) is an invaluable tool for visualizing subcellular structures in biomedical and life sciences. High-numerical-aperture (NA) immersion objective lenses are used to deliver excitation light to focus inside the biological tissue. The refractive index of tissue is commonly different from that of the immersion medium, which introduces spherical aberration, leading to signal and resolution degradation as imaging depth increases. However, the explicit dependence of this index mismatch-induced aberration on the involved physical parameters is not clear, especially its dependence on index mismatch. Here, from the vectorial equations for focusing through a planar interface between materials of mismatched refractive indices, we derive an approximate analytical expression for the spherical aberration. The analytical expression explicitly reveals the dependence of spherical aberration on index mismatch, imaging depth and excitation wavelength, from which we can expect the following qualitative behaviors: (1) Multiphoton signal and resolution degradation is less for longer excitation wavelength, (2) a longer wavelength tolerates a higher index mismatch, (3) a longer wavelength tolerates a larger imaging depth and (4) both signal and resolution degradations show the same dependence on imaging depth, regardless of NA or immersion on the condition that the integration angle is the same. Detailed numerical simulation results agree quite well with the above expectations based on the analytical approximation. These theoretical results suggest the use of long excitation wavelength to better suppress index mismatch-induced signal and resolution degradation in deep-tissue MPM.