  The 3D quasi-static particle-in-cell (PIC) algorithm is a very efficient
method for modeling short-pulse laser or relativistic charged particle
beam-plasma interactions. In this algorithm, the plasma response to a
non-evolving laser or particle beam is calculated using Maxwell's equations
based on the quasi-static approximate equations that exclude radiation. The
plasma fields are then used to advance the laser or beam forward using a large
time step. The algorithm is many orders of magnitude faster than a 3D fully
explicit relativistic electromagnetic PIC algorithm. It has been shown to be
capable to accurately model the evolution of lasers and particle beams in a
variety of scenarios. At the same time, an algorithm in which the fields,
currents and Maxwell equations are decomposed into azimuthal harmonics has been
shown to reduce the complexity of a 3D explicit PIC algorithm to that of a 2D
algorithm when the expansion is truncated while maintaining accuracy for
problems with near azimuthal symmetry. This hybrid algorithm uses a PIC
description in r-z and a gridless description in $\phi$. We describe a novel
method that combines the quasi-static and hybrid PIC methods. This algorithm
expands the fields, charge and current density into azimuthal harmonics. A set
of the quasi-static field equations are derived for each harmonic. The complex
amplitudes of the fields are then solved using the finite difference method.
The beam and plasma particles are advanced in Cartesian coordinates using the
total fields. Details on how this algorithm was implemented using a similar
workflow to an existing quasi-static code, QuickPIC, are presented. The new
code is called QPAD for QuickPIC with Azimuthal Decomposition. Benchmarks and
comparisons between a fully 3D explicit PIC code, a full 3D quasi-static code,
and the new quasi-static PIC code with azimuthal decomposition are also
presented.
