## 3 Results and Discussions
### Effects of Random Biological Aberrations  
In this section, we mainly focus on the effects of biological aberrations and the AO correction strategy to restore the quality of TPGB and TPBB. At first, it is necessary to understand the characteristics of biological aberrations and then simulate a realistic biological aberrations model. According to [PERSON] and [PERSON] et al. [21; 22], aberration caused by the variation of biological specimen refractive index can be considered random. Moreover, the amplitudes of higher-order Zernike modes are usually smaller than those of the lower-order ones. Furthermore, according to the experimental results of [PERSON] et al. [6; 7] on zebrafish embryo and mouse cortex, the first 65 Zernike modes are sufficient to describe biological aberrations. For superficial specimen (<0.1 mm), the aberration of biological sample is small, and RMS is approximately 0.1\(\lambda\). As the illumination position goes deeper, the aberration would increase gradually to even larger than 0.5\(\lambda\) (RMS). Thus, we use RMS to indicate the intensity of random aberrations caused by sample at different depths, and the variation ranges from 0.1\(\lambda\) to 1\(\lambda\). Based on the aberration models mentioned above, each aberration intensity randomly generates 100 wavefronts, and the coefficients' distribution of each Zernike mode is shown in Figure 8. The coefficients of tip, tilt and defocus modes are set to 0, since those modes only affect the position of the illumination beam and have no effects on the thickness. Figure 7: The illustration of the ‘self-reconstruction’ property of TPBB. The average TR of 100 groups of wavefronts is used to obtain the statistical results, as shown in Figure 9. For TPGB, the relationship between TR and RMS of biological aberration is nearly linear. For TPBB, the relationship is totally nonlinear. TR increases dramatically at first and then remains almost stable due to the'self-reconstruction' property. When RMS is less than 0.4\(\lambda\), the effect on the thickness of TPBB is stronger than that of TPGB, while the effect is far more serious for TPGB, as RMS changes from 0.5\(\lambda\) to 1.0\(\lambda\). This interesting result shows that the effect of random aberrations to TPBB is restrained by the'self-reconstruction' property. So, it would be better to use the fluorescent guide star (GS) excited by TPBB in the deeper specimen because the cross-section size of the GS is relatively smaller than that excited by TPGB, and the WFS can measure the wavefront more exactly. Of course, it should be a comprehensive consideration to choose which one as GS because the length of GS is another important factor. Figure 8: The coefficients’ distribution of each Zernike mode in the model of sample-induced aberrations, where the coefficients of tip, tilt and defocus modes are set to 0. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25 th and 75 th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers, and the outliers are plotted individually using the ‘+’ marker symbol. Figure 9: Relationship between TR and the random aberrations for TPBB and TPGB. Amplitude (RMS) is from 0.1\(\lambda\) to 1\(\lambda\). In order to determine how many Zernike modes are needed to correct to eliminate the distortion caused by biological aberrations, we set the coefficient of each Zernike mode to zero in turn to simulate the conventional AO correction process. The closer to 1 the TR values are, the better the correction performance is. The relationship between TR and the number of corrected Zernike modes under random aberration with variable RMS is shown in Figure 10. If we assume that the correction of the AO system is ideal when TR is smaller than 1.1, we find very similar results for TPBB and TPGB. For small aberrations (RMS \(\leq 0.1\lambda\)), the first 14 modes should be corrected to achieve ideal correction.