Quadrupole field in hairpin-type cathode gun and mechanism of axial-astigmatism production

Mechanism of axial astigmatism production in the hairpin-type cathode gun was investigated by numerical calculation. The electrostatic potential in the gun was first decomposed into the axially symmetric and quadrupole components, and then the trajectory behavior was investigated by the generalized trajectory method. The axial astigmatism of the hairpin-type cathode gun was found to depend strongly on the Wehnelt voltage; it is small as long as the Wehnelt voltage is close to cutoff condition but starts to increase rapidly when the drive voltage exceeds a certain threshold. The reason for a negligible axial astigmatism at or close to the cutoff lies in the mutual dependence of the tip electric field intensity and the quadrupole field strength near the cathode. The tip quadrupole field strength is given by the product of the tip field intensity and the difference in the cathode curvature in the two orthogonal cross sections. A negligible axial astigmatism of the hairpin-type cathode gun in usual operations is due to a small quadrupole field close to the cutoff condition where the tip field is weak.