Approach for simultaneous measurement of two-dimensional angular distribution of charged particles. III. Fine focusing of wide-angle beams in multiple lens systems

This paper provides a way of focusing wide-angle charged-particle beams in multiple lens systems. In previous papers [H. Matsuda et al., Phys. Rev. E 71, 066503 (2005); 74, 036501 (2006)], it was shown that an ellipsoidal mesh, combined with electrostatic lenses, enables correction of spherical aberration over wide acceptance angles up to $\pm 60°$. In this paper, practical situations where ordinary electron lenses are arranged behind the wide-angle electrostatic lenses are taken into account using ray tracing calculation. For practical realization of the wide-angle lens systems, the acceptance angle is set to $\pm 50°$. We note that the output beams of the wide-angle electrostatic lenses have somewhat large divergence angles which cause unacceptable or non-negligible spherical aberration in additional lenses. A solution to this problem is presented showing that lens combinations to cancel spherical aberration are available, whereby wide-angle charged-particle beams can be finely focused with considerably reduced divergence angles less than $\pm 5°$.

Figure 1

Electron trajectories in a typical Einzel lens (a) starting from a point on the plane P1 and (b) converging to a point on the plane P2. The electron energy is $1\mathrm{keV}$. The spherical aberrations $r_s$ at P1 and P2 relative to the radius $R$ of the lens are shown in each case, (c) and (d) corresponding to (a) and (b), respectively. Here $\theta$ is the angle of a trajectory at P1.

Figure 2

Generation and control of negative spherical aberration in the wide-angle Einzel-type lens with an acceptance angle of $\pm 50°$. (a) Electron trajectories with initial angles up to $\pm 50°$ in the focusing condition. Spherical aberrations $r_s$ at P1 are shown in the variations of (b) the ellipsoidal-shape parameter $\gamma$, (c) the voltage $V_2$ on the second electrode EL2, and (d) the distance $d$ between the object and the lens entrance. The solid line in each figure corresponds to the case of Fig. 2a.

Figure 3

Generation and control of negative spherical aberration in the wide-angle deceleration lens with an acceptance angle of $\pm 50°$. (a) Electron trajectories with initial angles up to $\pm 50°$ in the focusing condition. Spherical aberrations $r_s$ at P1 are shown in the variations of (b) the ellipsoidal shape parameter $\gamma$, (c) the voltage $V_2$ on the second electrode EL2, and (d) the distance $d$ between the object and the lens entrance. The solid line in each figure corresponds to the case of Fig. 3a.

Figure 4

Combination of the wide-angle Einzel-type lens and an ordinary Einzel lens. Electron trajectories with initial angles up to $\pm 50°$ are shown in two cases: (a) Simple lens combination in which spherical aberration is corrected at P1, and (b) fine lens combination in which spherical aberration is corrected at P2. Spherical aberrations $r_s$ at P1 and P2 are shown in different cases: (c) the case of Fig. 4a, (d) the case of optimizing $V_2$ and $\gamma$, which is the case of Fig. 4b, (e) the case of optimizing only $V_2$, and (f) the case of optimizing $V_2$ and $d$.

Figure 5

Combination of the wide-angle deceleration lens and an ordinary Einzel lens. Electron trajectories with initial angles up to $\pm 50°$ are shown in two cases: (a) Simple lens combination in which spherical aberration is corrected at P1, and (b) fine lens combination in which spherical aberration is corrected at P2. Spherical aberrations $r_s$ at P1 and P2 are shown in different cases: (c) the case of Fig. 5a, (d) the case of optimizing $V_2$ and $\gamma$, which is the case of Fig. 5b, (e) the case of optimizing only $V_2$, and (f) the case of optimizing $V_2$ and $d$.