Gravity compensation of bendable mirrors in Kirkpatrick-Baez systems with end-loss correction

Gravity-induced surface deviations in bendable mirrors degrade the performance of Kirkpatrick-Baez focusing systems at synchrotron and free-electron-laser beamlines. Conventional gravity compensation models ignore the inactive end length introduced by end clamping in four-roll benders. This omission leads to a systematic mismatch between predicted and measured surface shapes. This study develops an analytical model with end-loss correction within the Euler-Bernoulli beam framework for bendable mirrors. The model provides closed-form expressions for the required moments, the symmetric support forces, and the associated optimal dimensionless parameters. Finite element analysis verifies the predictions across representative geometries and load cases. With end loss included, the root-mean-square slope deviation decreases by about one order of magnitude to below 20 nrad, which meets the requirements for diffraction-limited focusing. This work establishes a robust framework for high-precision bendable mirror systems, with clear potential to reduce tuning cost and improve beamline optical performance.