Surface Parameter Estimation Method for Ultra-smooth Components Based on Generalized Beckmann-Kirchhoff Theory

In space-based gravitational wave detection systems， ultra-long laser interferometry arms are established between three communication satellites to detect the gravitational wave， requiring the space telescopes to perform both laser transmission and reception functions. However， the extreme weakness of gravitational wave signals imposes stringent requirements on optical path stability and backward stray light suppression in these telescopes， that is， the backward stray light must be less than 10-10 of the output laser. The primary source of backward stray light from space telescopes arises from surface scattering. Even in ultra-smooth optical elements （roughness less than 1 nm）， the surface scattering inevitably occurs during laser transmission， generating backward-scattered light that affects the gravitational wave detection. Therefore， precise characterization of ultra-smooth optical surfaces is critical for analyzing and suppressing telescope backward stray light levels. However， conventional non-contact measurement methods face significant challenges， including complex instrument structure， high experimental cost， and insufficient accuracy， making it difficult to achieve rapid and high-precision measurement of ultra-smooth optical surface. Consequently， there is an urgent need to develop a surface quality assessment method for ultra-smooth optical elements to meet the extraordinary precision requirements of gravitational wave space telescopes.For high-precision scattering measurement of ultra-smooth optical elements， Li B C's team proposed a method based on multi-channel cavity ring down technique. This method obtains the surface scattering rate by analyzing multi-channel ring-down signals， offering advantages including absolute measurement， immunity to laser source intensity fluctuation， and exceptional measurement accuracy. This work further combines the cavity ring-down-based scattering measurement technique with the Generalized Beckmann-Kirchhoff （GBK） scalar scattering model， to establish a method for estimating the surface parameters of ultra-smooth optical elements. In this method， the GBK scattering model is used to establish the scattering rate database of the optical elements， and the scattering rate of the ultra-smooth optical elements at different solid angles is measured based on the cavity ring down technique. Then， the scattering rate distribution measured by cavity ring down method is numerically fitted with the scattering rate database to obtain the surface parameters of the ultra-smooth optical elements， which provides a reference for the scattering characteristics analysis and stray light suppression of the ultra-smooth optical elements of space telescopes.The scattering rate distribution of ultra-smooth optical element measured by the cavity ring down technique was simulated and calculated， with the measured data numerically fitted through our proposed surface parameter estimation approach. As shown in Figure 3， the fitting achieves remarkable precision with an error of 0.16% and a goodness of fit of 0.999 8. Further evaluation of the method's performance across different parameters demonstrates that within the applicable range of GBK theory， the method can achieve an error of below 4% （typically0.999）. In addition， to address measurement errors induced by scattering overlap during bidirectional beam propagation in standing-wave cavity ring-down experiments， we analyzed key influencing factors. By integrating these findings with our predictive model， we quantified scattering overlap effects at various measurement angles and derived an optimized angular selection scheme to minimize interference.This work proposes a novel method for estimating surface parameters of ultra-smooth optical elements by combining GBK scalar scattering theory with cavity ring-down scattering measurement techniques. The simulation analysis shows that， within the scope of application of GBK scalar theory， the relative error of this method is less than 4%， and the fitting degree is greater than 0.96， confirming its high accuracy for rapid and precise measurement of surface quality parameters. In addition， aiming at the error caused by the scattering overlap of the back and forth propagation of the beam in the scattering measurement experiment of the standing wave cavity ring down method， we propose a measurement angle selection scheme considering both the signal-to-noise ratio and the measurement accuracy， so as to reduce the influence of the scattering overlap of the forward and backward propagation on the scattering measurement and subsequent surface quality estimation.