Experimental Reconstruction of Source 4D Phase Space Without Prior Knowledge of Transfer Matrix

TL;DR

This work addresses reconstructing the source transverse 4D phase space when the downstream transport is only approximately known. It introduces an aperture-scan method that leverages downstream 4D phase-space measurements and controlled emission-position displacements to determine a partial transfer matrix, which is completed using the symplectic condition and covariance evolution via Gauss-Newton fitting; the full source phase space is then recovered by inverting the transfer matrix. The method yields a source MTE of $69 \pm 2$ meV (and $73 \pm 2$ meV after PSF corrections) for a Na-K-Sb cathode, with low global position-momentum correlations and spatial features aligned to substrate fiducials, demonstrating physically consistent reconstruction. The approach is practical for existing accelerators with linear optics, enabling measurement of the transfer matrix and potentially guiding the design of “designer photocathodes” to tailor emission properties at microscopic scales.

Abstract

We experimentally demonstrate a method for reconstructing the transverse 4D phase space of an electron beam at the time of emission from downstream diagnostics of the 4D phase space. This method does not rely on detailed knowledge of the beamline transport, besides assuming that linearity and symplecticity are satisfied. We apply this method to measure the transverse position and momentum phase space of electrons emitted from a spatially-structured alkali-antimonide cathode. This method can uncover local correlations between emission location and momentum spread. We formulate this method analytically and investigate resolution limits.

Figures (7)

• Figure 1: A cartoon demonstrating the measurement of the partial transfer matrix. At the cathode (left), the beam is moved on the emission surface to positions $(x_L, 0)$ and $(0,y_L)$. At a downstream diagnostic (right), after evolving with the transfer matrix $\mathbf{M}$, the 4D phase space of the beam is measured and labeled by its centroids (indicated by stars). Each phase space is measured with the beam emitted at the correspondingly colored cathode position.
• Figure 2: a) Diagram of the aperture scan measurement. Beam (pink) is scanned over the aperture, and the transverse spatial distribution of the transmitted beam (blue) is imaged by the detector, which is converted into a transverse momentum distribution. b) Example of a measured 4D phase space, where a subset of the data is shown. Each image is a $(p_x,p_y)$ distribution measured when the beam is at the $(x,y)$ position on the aperture. c) The 4D phase space in b) projected into 2D planes along the $(x,p_x,y,p_y)$ axes.
• Figure 3: a) Measured 4D phase space of the beam at the aperture. Only a 1200 $\mu$m by 1200 $\mu$m part of the beam was measured. b) Reconstructed source 4D phase space of the beam at the cathode.
• Figure 4: a) Detailed image of the reconstructed source $x$-$y$ position phase space. Linecut profiles are indicated by the dashed lines and correspondingly colored with the displayed profiles in c) and d). b) Scanning electron microscope image of the substrate. Dark regions correspond to SiO$_2$, and copper for light regions. c) Horizontal linecut profiles, taken on (orange) and off (green) the features in a). The off-feature linecut is treated as background and subtracted from the on-feature linecut, yielding the linecut in blue. d) Vertical linecut profile. Black dashed lines in c) and d) indicate the position of features corresponding to known QE modulations induced by the substrate pattern.
• Figure 5: a) 2D projections of a 4D phase space of the beam measured at beamline optics settings where the intensity modulations in the $x$-$y$ phase space are significantly distorted. b) Source 4D phase space reconstructed from a). In the source spatial distribution, the dark circular features within the red circles correspond with fiducial markers on the cathode.