Chromatic and Spherical Aberration Correction with Hexapole and Quadrupole Fields

We report the development of a chromatic and spherical aberration corrector based on combinations of hexapole and quadrupole fields. Thick hexapole fields are used to generate negative third order spherical aberration and to correct residual axial and off-axial aberrations. As an alternative to the use of round transfer lenses placed between the hexapoles, a quadrupole multiplet producing superimposed electric and magnetic quadrupole fields is used to produce negative chromatic aberration. This quadrupole multiplet also functions as a transfer doublet within the corrector. The simultaneous correction of chromatic and spherical aberrations using this corrector design is described, and a resolution improvement is demonstrated for cases where the energy spread is limiting.

💡 Research Summary

The paper presents a novel electron‑optical corrector that simultaneously compensates chromatic (Cc) and spherical (Cs) aberrations by integrating thick hexapole fields with a super‑imposed electric‑magnetic quadrupole multiplet, termed the hexapole‑quadrupole (HQ) corrector. Traditional round objective lenses suffer from inherent third‑order spherical aberration (C3,0) and first‑order chromatic aberrations (Cc1,1,0 and Cc1,1,2), which limit resolution especially at low accelerating voltages or when inelastic scattering broadens the energy distribution. Existing solutions either use hexapole‑only spherical correctors or quadrupole‑octupole (QO) chromatic correctors; however, combining them in series introduces additional geometric aberrations and mechanical complexity.

In the HQ design, two thick hexapoles (H1, H2) generate a negative C3,0 by exploiting the primary three‑fold astigmatism inherent to a thick hexapole field. Instead of the conventional round‑lens transfer doublet placed between the hexapoles, a quadrupole multiplet functions as the transfer optic. By superimposing electric and magnetic quadrupole fields with opposite polarity, the multiplet produces negative first‑order chromatic coefficients Cc1,1,0 (defocus) and Cc1,1,2 (astigmatism). The inner poles (Q2, Q3, Q6, Q7) host an elliptical (line‑focus) beam where the strongest chromatic correction occurs, while the outer poles (Q1, Q4, Q5, Q8) provide supplementary correction.

A key element is the “Russian quadruplet” – an antisymmetric arrangement of four thin quadrupoles (D‑C‑D‑C in x, C‑D‑C‑D in y). Using ray‑transfer matrix analysis, the authors derive conditions under which this quadruplet forms a conjugate plane with unit magnification and 180° rotational symmetry. In practice, thick quadrupoles are required to achieve sufficient field strength; therefore, the authors extend the matrix formalism to include quadrupole thickness (t) and field strength (α), solving the resulting equations numerically. The distance d1 between the entrance of the first quadrupole and the conjugate plane is a critical design parameter, as it determines whether the hexapoles can be positioned without mechanical interference.

Higher‑order aberrations generated by the interaction of hexapole and quadrupole fields (second‑order coma, fourth‑order coma, five‑fold astigmatism) are mitigated by employing two symmetric quadrupole quadruplets and inserting a magnetic multipole at the central conjugate plane. Additional windings on the poles generate dipole, hexapole, and octupole components for fine‑tuning of residual aberrations.

Experimental validation was performed on a 200 kV transmission electron microscope equipped with the HQ corrector (overall height ≈ 72 cm). The required electrode voltages scale with accelerating voltage (≈ 2.5× higher at 200 kV, ≈ 4.6× at 300 kV compared with 100 kV). Chromatic coefficients were measured by analyzing the energy‑dependent defocus and astigmatism extracted from power spectra of images taken at different beam energies. Without correction, Cc was 1.1 mm (≈ 6.5 nm/eV). After alignment and activation of the quadrupole multiplet, both Cc1,1,0 and Cc1,1,2 were reduced to ≈ 0.01 mm (≈ 0.06 nm/eV). Spherical aberration C3,0 of the objective lens (initially 0.7 mm) was also driven to near‑zero by energizing the hexapoles. Residual second‑order chromatic terms (Cc2) remained larger than the second‑order chromatic defocus, but their impact is minor once the first‑order terms are corrected.

The HQ corrector thus achieves simultaneous chromatic and spherical correction with fewer components than a conventional hexapole‑plus‑QO cascade, reduces mechanical complexity, and offers flexibility across a wide voltage range. The authors note that while higher‑order chromatic terms are not yet fully compensated, the dominant first‑order corrections already yield a substantial resolution gain, particularly for low‑voltage imaging of beam‑sensitive specimens. Future work is suggested to integrate octupole and dodecapole windings for complete high‑order correction and to develop automated alignment algorithms for routine operation.