## 1 Introduction  
The two-photon excitation microscope (TPM) has received much attention and has been rapidly developed since being proposed by [PERSON] et al. in 1990 [1]. Compared with conventional confocal microscopy, TPM provides high spatial resolution imaging with lower amounts of scattering in deep regions of biological samples because of longer wavelength illumination and nonlinear excitation effects. However, the temporal resolution of TPM is limited, since a scanning system is indispensable in order to obtain a large field of view (FOV). In recent years, the Bessel beam (a kind of diffraction-free beam) has drawn much attention to be used as an illumination beam of TPM, because it can achieve high spatiotemporal resolution while maintaining large FOV. In 2013, [PERSON] et al. [2] developed a two-photon Bessel light-sheet microscope, and the light-sheet penetration depth was increased by a factor of 3-5 compared to linear excitation with a Gaussian beam. In 2020, [PERSON] et al. [3] developed a Bessel two-photon laser scanning microscope for high-speed volumetric fluorescence imaging of neurovascular dynamics. The 22-times increase in the axial excitation range allowed them to use 2D scanning of the Bessel beam to image a volume, instead of the 3D scanning of the Gaussian beam.  
Although using two-photon Bessel beam (TPBB) as illumination beam could provide many advantages, the performance of the microscopy system (especially resolution) is still dramatically affected by the sample-induced optical aberrations, as the illumination beam goes deep into the biological tissues [4]. Especially when imaging inside semi-transparent samples, the image quality is weakly affected by scattering up to a depth of several scattering lengths [5], so optical aberrations would be the main influence factor. Optical aberrations will lead to a spreading of the focusing spot inside the sample, both in axial and lateral directions, which will increase the thickness of TPBB and ultimately reduce the resolution of the microscope.  
Adaptive optics (AO) is the most powerful technique to correct the aberrations of optical systems and has been widely used in Gauss beam illuminated two-photon microscopes [6; 7; 8] (TPGM) and light-sheet microscopes [9; 10; 11] for the past few years. However, there has been little research on integrating the AO elements in a two-photon Bessel microscope (TPBM) up to now. Because of the differences in principles [12], methods [13] and light path conjugation relationships [14] to generate the TPBB and two-photon Gaussian beam (TPGB), the sample-induced aberrations have totally different effects even if the illumination beam passes the same part of biological sample. Therefore, quantitative analysis of how aberrations affect the TPBB is rather important to design a reasonable AO system for TPBM.  
In this paper, we analyzed the effects of optical aberrations to the thickness of TPGB and TPBB by simulating the practical optical system and imaging process. We found that the thickness variations of these two beams are both related to the azimuthal order of Zernike modes but have different effect models. TPGB is more sensitive to those Zernike modes with lower azimuthal order, while TPBB is more sensitive to the higher azimuthal-order Zernike modes. Moreover, it is necessary to design a new strategy to correct the aberrations according to the effects of different Zernike modes in order to maximize the correction capability of the corrector and reduce the correction errors for those insensitive Zernike modes. Finally, further reasonable advice on AO is provided.