A discrete square global grid system based on the parallels plane projection

Knowledge Innovation Project of Chinese Academy of Sciences [KZCX3-SW-347]; National Science Fund for Distinguished Young Scholar [40225004]

Parallels plane projection and its geometric features

A new equivalent map projection called the parallels plane projection is proposed in this paper. The transverse axis of the parallels plane projection is the expansion of the equator and its vertical axis equals half the length of the central meridian. On the parallels plane projection, meridians are projected as sine curves and parallels are a series of straight, parallel lines. No distortion of length occurs along the central meridian or on any parallels of this projection. Angular distortion and the proportion of length along meridians (except the central meridian) introduced by the projection transformation increase with increasing longitude and latitude. A potential application of the parallels plane projection is that it can provide an efficient projection transformation for global discrete grid systems.

A discrete square global grid system based on the parallels plane projection

We developed a direct partitioning method to construct a seamless discrete global grid system (DGGS) with any resolution based on a two-dimensional projected plane and the earth ellipsoid. This DGGS is composed of congruent square grids over the projected plane and irregular ellipsoidal quadrilaterals on the ellipsoidal surface. A new equal area projection named the parallels plane (PP) projection derived from the expansion of the central meridian and parallels has been employed to perform the transformation between the planar squares and the corresponding ellipsoidal grids. The horizontal sides of the grids are parts of the parallel circles and the vertical sides are complex ellipsoidal curves, which can be obtained by the inverse expression of the PP projection. The partition strategies, transformation equations, geometric characteristics and distortions for this DGGS have been discussed. Our analysis proves that the DGGS is area-preserving while length distortions only occur on the vertical sides off the central meridian. Angular and length distortions positively correlate to the increase in latitudes and the spanning of longitudes away from a chosen central meridian. This direct partition only generates a small number of broken grids that can be treated individually.

Coma measurement of projection optics in lithographic tools based on relative image displacements at multiple illumination settings

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. (c) 2007 Optical Society of America.

Distortion of beam shape by a rotating double-prism wide-angle laser beam scanner

We describe the rigorous results of a wide-angle laser beam scanner, obtained with the help of the vector refraction theory. Using the rigorous results, the distortion of the beam shape is discussed. The distortion to the beam varies with the different relative angles of double prisms. The scanner expands the beam in some directions while it contracts the beam in other directions. According to the conservation of energy, the distribution of the laser intensity is changed as well. (c) 2006 Society of Photo-Optical Instrumentation Engineers.

Method for measuring the lateral aberrations of a lithographic projection system with mirror-symmetric FOCAL marks

A novel method for measuring the imaging quality of a projection system with mirror-symmetric FOCAL marks is proposed, and the principle of the method is described. Through experiments, it is demonstrated that not only the axial aberrations but also the lateral aberrations can be measured with high accuracy by the method. The advantages of the method include obtaining more aberrations than the FOCAL technique and making it much simpler to perform a full-scale measurement of the imaging quality of a lithographic projection system. (C) 2006 Society of Photo-Optical Instrumentation Engineers.

Aberration measurement of projection optics in lithographic tools based on two-beam interference theory

The degradation of image quality caused by aberrations of projection optics in lithographic tools is a serious problem in optical lithography. We propose what we believe to be a novel technique for measuring aberrations of projection optics based on two-beam interference theory. By utilizing the partial coherent imaging theory, a novel model that accurately characterizes the relative image displacement of a fine grating pattern to a large pattern induced by aberrations is derived. Both even and odd aberrations are extracted independently from the relative image displacements of the printed patterns by two-beam interference imaging of the zeroth and positive first orders. The simulation results show that by using this technique we can measure the aberrations present in the lithographic tool with higher accuracy. (c) 2006 Optical Society of America.

Aberration measurement of projection optics in lithographic tools by use of an alternating phase-shifting mask

As a critical dimension shrinks, the degradation in image quality caused by wavefront aberrations of projection optics in lithographic tools becomes a serious problem. It is necessary to establish a technique for a fast and accurate in situ aberration measurement. We introduce what we believe to be a novel technique for characterizing the aberrations of projection optics by using an alternating phase-shifting mask. The even aberrations, such as spherical aberration and astigmatism, and the odd aberrations, such as coma, are extracted from focus shifts and image displacements of the phase-shifted pattern, respectively. The focus shifts and the image displacements are measured by a transmission image sensor. The simulation results show that, compared with the accuracy of the previous straightforward measurement technique, the accuracy of the coma measurement increases by more than 30% and the accuracy of the spherical-aberration measurement increases by approximately 20%. (c) 2006 Optical Society of America.

Measurement technique for in situ characterizing aberrations of projection optics in lithographic tools

As the feature size decreases, degradation of image quality caused by wavefront aberrations of projection optics in lithographic tools has become a serious problem in the low-k1 process. We propose a novel measurement technique for in situ characterizing aberrations of projection optics in lithographic tools. Considering the impact of the partial coherence illumination, we introduce a novel algorithm that accurately describes the pattern displacement and focus shift induced by aberrations. Employing the algorithm, the measurement condition is extended from three-beam interference to two-, three-, and hybrid-beam interferences. The experiments are performed to measure the aberrations of projection optics in an ArF scanner. (C) 2006 Optical Society of America.

Method for measuring the lateral aberrations of a lithographic projection system with mirror-symmetric FOCAL marks

A novel method for measuring the imaging quality of a projection system with mirror-symmetric FOCAL marks is proposed, and the principle of the method is described. Through experiments, it is demonstrated that not only the axial aberrations but also the lateral aberrations can be measured with high accuracy by the method. The advantages of the method include obtaining more aberrations than the FOCAL technique and making it much simpler to perform a full-scale measurement of the imaging quality of a lithographic projection system. (C) 2006 Society of Photo-Optical Instrumentation Engineers.

A novel method to measure coma aberration of projection system

A novel method to measure coma aberration by pattern displacements at different defocus positions is proposed in this paper. The effect of defocus on coma-induced pattern displacement is analyzed. The measuring principle of the method is described in detail. Using the simulation program PROLITH, the proportionality factors between pattern displacement and coma aberration at different defocus positions are calculated. It is proved that the method is simple to perform and the measurement accuracy of coma can increase approximately by 25% by this novel method. (c) 2006 Elsevier GmbH. All rights reserved.