  The multi-bend achromat (MBA), which often serves as a building block for
modern low-emittance storage rings, is composed of a repetition of unit cells
with optimized optical functions for low emittance in the achromat center, as
well as end cells for dispersion and optics matching to insertion devices.
  In this work, we describe the simplest stable class of unit cells that are
based on a longitudinal Fourier expansion, transforming Hill equations to
Mathieu equations. The resulting cell class exhibits continuously changing
dipolar and quadrupolar moments along the beam path. Although this elementary
model is defined by only three parameters, it captures a significant amount of
notions that are applied in the design of MBAs. This is especially interesting
as Mathieu cells can be viewed as an elementary extension of Christofilos'
original model of alternating-gradient focusing, while their sinusoidal bending
and focusing functions lend themselves to future applications in undulator-like
structures.
  Mathieu cells can be used to estimate the range of reasonable cell tunes and
put an emphasis on the combination of longitudinal gradient bending and reverse
bending, as well as on strong horizontal focusing to reach emittances lower
than the classic theoretical minimum emittance cell. Furthermore, the lowest
emittances in this model are accompanied by small absolute momentum compaction
factors.
