## 3 Results and Discussions  
### Effects of Single Zernike Mode Aberration  
The first 104 Zernike modes and their distortion effects along the direction of illumination are simulated. For each mode, a series of amplitudes from 0.1\(\lambda\) to 2\(\lambda\) (PV) are simulated. For simplicity, only the first 14 Zernike modes with 1.0\(\lambda\) (PV) and the distorted PSF in XY plane are shown in Figure 3. Figure 4 gives the effect on TR of different Zernike modes with four sets of PVs (0.3\(\lambda\), 0.8\(\lambda\), 1.3\(\lambda\) and 1.8\(\lambda\)). According to these results, we can summarize the general rules of the aberrations' effect on TPGB and TPBB. For both TPGB and TPBB, as the amplitude of aberration increases, the TR becomes larger, i.e., the effect becomes more serious. The effects on TR of lower-order modes are greater than that of higher-order modes under relatively small amplitude (TPGB: PV \(<\) 1.3\(\lambda\); TPBB: PV \(<\) 0.8\(\lambda\)). For the higher-order Zernike modes, the TPBB's TR would reach to a rather high level under smaller amplitude compared to TPGB. What is more interesting is that the Zernike modes with the same azimuthal order have different effects on the thickness of TPGB and TPBB, as shown in Figure 5. Generally, the Zernike modes with lower azimuthal orders (m \(\leq\) 4) have relatively stronger effects on the thickness of TPGB, while they have smaller effects on thickness of TPBB. In contrast, the Zernike modes with higher azimuthal orders (n \(-\) 1 m \(\leq\) 4) have relatively stronger effects on the thickness of TPBB, while they have smaller effects on the thickness of TPGB. In order to explain this difference, we go back to the definition of Zernike modes. According to Equation (3), there are fewer lower-order terms of r in R\({}_{\rm n}^{\rm m}(r)\) as the increasing of azimuthal order m. Figure 6 shows how a radial circle polynomial R\({}_{\rm n}^{\rm m}(r)\) varies with r and m. When the radial order is odd, the fluctuation of the aberrated wavefront gradually shifts from the center to the edge as the azimuthal order increases. The same thing happens when radial order is even; however, the central fluctuation is most dramatic when the azimuthal order is 2 rather than 0. The intensity profile of TPGB and TPBB at the plane conjugated to the WFS are solid circle and hollow ring, respectively. Therefore, TPGB is more sensitive to the central fluctuation of the wavefront, and TPBB is more sensitive to the fluctuation in the periphery of the wavefront. Thus, TPGB is more sensitive to the Zernike modes whose azimuthal orders are relatively lower, especially less than 4. In contrast, those Zernike modes whose difference between radial order and azimuthal order is less than 4 have greater influence on the thickness of TPBB. Figure 5: (**a**) TPGB and (**b**) TPBB are sensitive to the Zernike modes with different azimuthal orders. Amplitude (PV) of each mode is 1.6\(\lambda\). Another point worth noting about the Bessel beam is its'self-reconstruction' property [20]. As shown in Figure 7, TPBB will gradually restore to its original shape as the aberration amplitude increases. Generally, the'self-reconstruction' process takes place when the amplitude changes from 0.8\(\lambda\) to 1.5\(\lambda\). For the Zernike modes that TPBB is sensitive to, at first TR is stabilized at about 4, and then it continues to increase after 1.6\(\lambda\). For other Zernike modes, TR increases first and then declines as TPBB gradually reverts to its original shape. That is to say, the growth curve of TR is not monotonic but fluctuant. According to the above results, researchers in the field of AO should pay more attention to the different affection models of Zernike aberration to TPGB and TPBB. Considering all Zernike modes with the same weights would be a waste of the correction capability of the correction devices.