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

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. Martínez & 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.