Analysis of the transverse phase space diagnostic in laser wakefield accelerators using the principal planes

The focus-scan scheme has been adopted to diagnose the emittance of the electron beams produced from laser wakefield accelerators. The spot size of the electron beam has an energy dependence after propagating through a beamline consisting of two or more quadrupoles, which can be used to fit the beam emittance and the Courant-Snyder (C-S) parameters. We use the concept of the principal planes to unify the analysis of the focus-scan scheme. The fitting procedures and the effects of the length variation between the source, magnets, and screen on the diagnostic are presented. Particle-in-cell simulations using plasmas with different downramp lengths are conducted to study their effects on the measurements of the emittance and C-S parameters. Our analysis provides a unified understanding of the focus-scan scheme and can be applied to beamlines consisting of more quadrupoles.