Third‐order aberration theory of Wien filters for monochromators and aberration correctors

Abstract

Third-order aberrations at the first and the second focus planes of double focus Wien filters are derived in terms of the following electric and magnetic field components--dipole: E1, B1; quadrupole: E2, B2; hexapole: E3, B3 and octupole: E4, B4. The aberration coefficients are expressed under the second-order geometrical aberration free conditions of E2 = -(m + 2)E1/8R, B2 = -mB1/8R and E3R2/E1 - B3R2/B1 = m/16, where m is an arbitrary value common to all equations. Aberration figures under the conditions of zero x- and y-axes values show very small probe size and similar patterns to those obtained using a previous numerical simulation [G. Martinez & K. Tsuno (2004) Ultramicroscopy, 100, 105-114]. Round beam conditions are obtained when B3 = 5m2B1/144R2 and (E4/E1 - B4/B1)R3 = -29m2/1152. In this special case, aberration figures contain only chromatic and aperture aberrations at the second focus. The chromatic aberrations become zero when m = 2 and aperture aberrations become zero when m = 1.101 and 10.899 at the second focus. Negative chromatic aberrations are obtained when m < 2 and negative aperture aberrations for m < 1.101. The Wien filter functions not only as a monochromator but also as a corrector of both chromatic and aperture aberrations. There are two advantages in using a Wien filter aberration corrector. First, there is the simplicity that derives from it being a single component aberration correction system. Secondly, the aberration in the off-axis region varies very little from the on-axis figures. These characteristics make the corrector very easy to operate.