Ke Liu

Multilateral shearing interferometers (multi-LSIs) utilize the phase difference information of multiple shear directions with high accuracy and strong noise resistance. However, the interferogram fringe contrast of multi-LSIs can reverse due to the effect of contrast modulation, leading to incorrect measurement result. This issue has lacked comprehensive quantitative research to effectively guide the elimination of its effects. In this work, a quantitative analysis of the contrast modulation is proposed. This paper rigorously derives contrast modulation function, analyzing the effect of aberration and shear ratio on it. It is the first time that a quantitative theoretical model of aberration measurement range and shear ratio has been established, offering guidance in practical application for precise shear ratio adjustment to eliminate modulation effect and extend measurement range. By eliminating modulation effects, phase difference information from multiple directions can be effectively utilized, maximizing the benefits of multi-LSIs. In simulation and experiment, the distribution of fringe contrast reversal bands and the variation of aberration measurement range with shear ratio are consistent with theoretical analysis, which verifies the validity of the proposed approach. This approach aims to address the issue of interferogram fringe contrast inversion caused by modulation in multi-LSIs, providing a theoretical basis and practical guidance for precise measurement.

The current research on quadriwave lateral shearing interferometry (LSI) is primarily focused on the measurements of plane wave or quasiplane wave. When directly measuring spherical waves with high numerical apertures (NA>0.35), it will generate additional systematic errors, which affect measurement accuracy. To make the quadriwave LSI applicable to the measurements of a high-NA spherical wave, this paper proposes a fast and direct calibration method by establishing the geometry model of the quadriwave LSI with high-NA spherical wave incidence. The expression for the optical path difference represented systematic errors introduced by high-NA spherical waves in the shearing wavefronts are derived, which utilize a ninth-order Taylor expansion and Zernike polynomials fitting to achieve the high accuracy required by the high-NA spherical wave incidence. Then, the systematic errors are directly calibrated in the shearing wavefronts of four directions. This paper presents the theoretical analysis and verifies the feasibility and reliability of the proposed method through simulations and experiments, achieving a good measurement accuracy.

The wavefront reconstruction is a crucial step in determining the performance of wavefront detection instruments. The wavefront reconstruction algorithm is primarily evaluated in three dimensions: accuracy, speed, and noise immunity. In this paper, we propose a hybrid zonal reconstruction algorithm that introduces slope and curvature information in the diagonal, anti-diagonal, horizontal, and vertical directions by dividing the neighbor sampling points into subregions in groups of four. By canceling the same parameters in integration equations, an algorithm using multi-directional slope–curvature information is achieved with only two sets of integration equations in each subregion, reducing the processing time. Simulation experiments show that the relative root-mean-square reconstruction error of this algorithm is improved by about 4 orders of magnitude compared with existing algorithms that use multi-directional slope information or slope–curvature information alone. Compared with the hybrid multi-directional slope–curvature algorithm, the proposed algorithm can reduce computation time by about 50% as well as provide better noise immunity and reconstruction accuracy. Finally, the validity of the proposed algorithm is verified by the null test experiment.

A new method, to the best of our knowledge, based on double-slit (DS) interference is proposed to accurately estimate the shear ratio of the system, with plane wave or spherical wave incidence. Existing shear ratio calibration methods, designed primarily for lateral shearing interferometry (LSI) with plane wave incidence, are not applicable to LSIs directly testing divergent or convergent spherical waves. Equations for calculating the shear ratio using the fringe spacing of the DS interferogram and the NA of the incident spherical wave are derived in this paper. The simulation result shows that the relative error of the shear ratio value is about 0.3%, when the shear ratio is 0.1. In the experiment, the quadriwave LSI is designed with a plug-in feature. The shear ratio at integer multiples of 1/6 Talbot distance from the modified Hartmann mask was calibrated using a DS, and the results were in good agreement with theoretical values, confirming the accuracy of the method. Subsequently, with the assistance of an inductance micrometer, the shear ratio was calibrated at intervals of 0.5 mm, and the results closely matched the theoretical variation of the shear ratio caused by displacement, confirming the high precision of the method.

Channeled spectropolarimeter (CSP) measures spectrally resolved Stokes vector of light and Mueller matrix of sample from a snapshot. While reconstruction and calibration methods for Stokes CSP have been well established, their Mueller CSP counterparts are lacking. In this paper, we propose methods for Mueller spectrum reconstruction and Mueller CSP calibration. Mueller CSP is modeled as a modulation matrix, linking the Mueller spectrum to be measured and the modulated spectrum from the spectrometer. We describe an optimization problem to solve the Mueller spectrum, where both the regularizer and the residual threshold constrain the result, making our reconstruction accurate, efficient, and noise-robust. The Stokes spectrum generated by polarization state generator and the analyzing vector of polarization state analyzer are measured in situ, the convolution of which construct the calibrated modulation matrix of Mueller CSP. Total polarimetric errors and spectroscopic errors are treated as a whole and represented by the calibrated modulation matrix. Both imaging and non-imaging Mueller CSP are experimentally calibrated. Reconstruction results show high accuracy with a root-mean-square error (RMSE) of 0.0371. The proposed methods help make Mueller CSP practical and have the potential to be general reconstruction and calibration methods for imaging and non-imaging Stokes-Mueller CSP.

An efficient calibration method of total polarimetric errors in a channeled spectropolarimeter (CSP) is proposed and experimentally demonstrated. Total polarimetric errors, including alignment and retardance errors as well as those caused by nonideal retarders and the polarizer in CSP, are considered and calibrated. We first construct the calibrated modulation matrix of CSP by directly measuring the Mueller matrix spectrum of the polarization module in CSP. Compared to previously reported calibration works that required 1074 measurements, our calibration requires only 16 individual measurements, which reduces the measurement time by 67-fold while ensuring high accuracy with a maximum rms error less than 0.02. Further experimental test on three types of different CSP systems confirms the efficiency, reliability, and accuracy of the proposed calibration method.

The illumination system design for high numerical aperture (NA) anamorphic objectives is a key challenge for extreme ultraviolet lithography. In this paper, a reverse design method of the off-axis mixed-conic-surface-type relay system and an automatic arrangement method of field facets are proposed to design a high NA anamorphic illumination system. The two off-axis relay mirrors are fitted into different conic surfaces based on the conjugation of the mask plane and field facet and that of the illumination exit pupil and pupil facet. To eliminate ray obscuration between neighboring field facets, the field facets are automatically arranged according to the distances that are determined by the relative tilt angles of neighboring field facets under the current illumination mode. The proposed methods are applied in the design of an illumination system matching the NA0.60 anamorphic objective. Simulation results show that the uniformity of the scanning energy distribution can reach 99% on the mask plane under different illumination modes.

Optical vortex beams (VB) carrying orbital angular momentum is of great importance in many fields. In this letter, combining a polarization modulation system and high NA lens, we propose a new method for detecting the topological charge (TC) value of VB. According to the focal field intensity distribution of VB after polarization modulation, we explore the correlation between the intensity and the TC of VB. The results reveal that the tightly focusing intensity represents a smallest solid spot when the TC value of VB equals the polarization azimuthal index. Based on this property, we can determine arbitrary states of VB by the focal intensity, especially in the case of higher charge.