  In previous studies of spin-to-orbital angular momentum (AM) conversion in
laser high harmonic generation (HHG) using a plasma target, one unit of spin AM
is always converted into precisely one unit of OAM [1, 2]. Here we show,
through analytic theory and numerical simulations, that we can exchange one
unit of SAM for a tuneable amount of OAM per harmonic step, via the use of a
structured plasma target. In the process, we introduce a novel framework to
study laser harmonic generation via recasting it as a beat wave process. This
framework enables us to easily calculate and visualise harmonic progressions,
unify the "photon counting" and "symmetry-based" approaches to HHG and provide
new explanations for existing HHG results. Our framework also includes a
specific way to analyse simultaneously the frequency, spin and OAM content of
the harmonic radiation which provides enhanced insight into this process. The
prospects of using our new framework to design HHG configurations with tuneable
high-order transverse modes, also covering the design of structured plasma
targets, will be discussed.
