The correlation between the coma sensitivity of the alternating phase-shifting mask (Alt-PSM) mark and the mark’s structure is studied based on the Hopkins theory of partially coherent imaging and positive resist optical lithography (PROLITH) simulation. It is found that an optimized Alt-PSM mark with its phase width being two-thirds its pitch has a higher sensitivity to coma than Alt-PSM marks with the same pitch and the different phase widths. The pitch of the Alt-PSM mark is also optimized by PROLITH simulation, and the structure of $p=1.92\lambda /\mathrm{NA}$ and $pw=2p/3$ proves to be with the highest sensitivity. The optimized Alt-PSM mark is used as a measurement mark to retrieve coma aberration from the projection optics in lithographic tools. In comparison with an ordinary Alt-PSM mark with its phase width being a half its pitch, the measurement accuracies of $Z_7$ and $Z_{14}$ apparently increase.

In this paper, we propose a novel method for measuring the coma aberrations of lithographic projection optics based on relative image displacements at multiple illumination settings. The measurement accuracy of coma can be improved because the phase-shifting gratings are more sensitive to the aberrations than the binary gratings used in the TAMIS technique, and the impact of distortion on displacements of aerial image can be eliminated when the relative image displacements are measured. The PROLITH simulation results show that, the measurement accuracy of coma increases by more than 25% under conventional illumination, and the measurement accuracy of primary coma increases by more than 20% under annular illumination, compared with the TAMIS technique.

This paper aims to solve the consensus problem of three different types of multi-agent systems under fixed communication delay. For these three different types of multi-agent systems, three new control protocols are constructed to solve the consensus issue of multi-agent systems with coupling weights, which is rarely considered in other related articles. For fixed coupled multi-agent systems, a new control strategy is designed by Lyapunov theory, matrix theory, and the inequality method to ensure the consensus of multi-agent systems. An adaptive coupling weight updating scheme is proposed for adaptive coupling multi-agent systems to achieve the consensus state of the multi-agent systems. Finally, experimental results show the effectiveness of the proposed three different algorithms.

When propagating in free space, the transversal distribution of the degree of polarization of an anisotropic electromagnetic Gaussian–Schell model (AEGSM) beam will generally undergo a complex evolution process. We find that this transversal distribution of the degree of polarization of an AEGSM beam can be controlled by exploiting the partial correlation properties of the source. The main research of our paper falls into two parts. First, the concept of analogical propagation of the transversal distribution of the degree of polarization is proposed, and the condition for an AEGSM beam having an analogical propagation is obtained. When an AEGSM beam is on analogical propagation, the distribution of the degree of polarization on any cross section of the beam is always similar to that on the source plane, except that the size of the distribution pattern will expand continuously as the propagation distance increases. Second, the far-field transversal distribution of the degree of polarization is considered, and the condition for the far-field transversal polarization distribution of an AEGSM beam to be always of circularly symmetric shape, no matter how complicated it is on the source, is obtained. Our research is expected to find applications in areas that make use of the polarization properties of random electromagnetic beams.

We discuss the aberration-induced intensity imbalance of alternating phase-shifting mask (Alt-PSM) in lithographic imaging, in contrast to numerous studies on mask-induced intensity imbalance. Based on the Hopkins theory of partial coherent imaging, a linear relationship between the intensity difference of adjacent peaks in an Alt-PSM image and even aberration is established by formulations and verified by numerical results. The application of the linear relationship is briefly discussed.

We propose an in situ technique for measuring an even aberration of lithographic projection optics. By using the Hopkins theory of partially coherent imaging and the thick-mask model, the linear relationship between the intensity difference of adjacent peaks in an alternating phase-shifting mask image and an even aberration is established by equations and verified by numerical results. The sensitivity of measuring the even aberration of lithographic projection optics based on this linear relationship is analyzed, and the measurement mark is designed accordingly. Measurement performance of the present technique is evaluated using the lithographic simulator PROLITH, which shows that the present technique is capable of measuring the even aberration of lithographic projection optics with ultrahigh measurement accuracy.