Monte Carlo investigation of spatiotemporal distortions in attosecond soft X-ray pulse focusing using a two-stage toroidal mirror system

Attosecond soft X-ray pulses provide excellent temporal and spatial coherence, enabling ultrafast probing of electronic and structural dynamics. Fully exploiting their capabilities requires high demagnification to achieve the necessary power density, but such conditions induce strong spatiotemporal coupling distortions. In this work, we develop a Geant4-based Monte Carlo framework for quantitatively evaluating these distortions during attosecond soft X-ray focusing. Aberration-theory analysis and simulations of a single toroidal mirror reveal that coma increases rapidly with the demagnification ratio, fundamentally limiting single-stage performance. To overcome this constraint, we propose a two-stage toroidal focusing architecture. Using analytically derived structural parameters as initial conditions, the system is numerically optimized within Geant4. The final design achieves a demagnification of 15.5×, producing a focal-spot full width at half maximum (FWHM) of 14.4 μm, while reducing temporal broadening within the spatial FWHM to 153.0 as peak-to-valley (PV) and 18.0 as root-mean-square (RMS). To enhance practical relevance, we further incorporate independent meridional and sagittal slope errors for both toroidal mirrors and quantify the resulting degradations of spatial focusing and temporal broadening. These results demonstrate that the two-stage configuration effectively suppresses coma and minimizes spatiotemporal distortions, providing a robust approach for high-precision attosecond soft X-ray beamline development.