In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs...In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs,are in big demand due to their optimality properties important for many applications.The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings.This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems.Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems,and O(k)complexity estimation is provided for a problem with k generators.展开更多
Laser interference lithography is an attractive method for the fabrication of a large-area two-dimensional planar scale grating,which can be employed as a scale for multi-axis optical encoders or a diffractive optical...Laser interference lithography is an attractive method for the fabrication of a large-area two-dimensional planar scale grating,which can be employed as a scale for multi-axis optical encoders or a diffractive optical element in many types of optical sensors.Especially,optical configurations such as Lloyd's mirror interferometer based on the division of wavefront method can generate interference fringe fields for the patterning of grating pattern structures at a single exposure in a stable manner.For the fabrication of a two-dimensional scale grating to be used in a planar/surface encoder,an orthogonal two-axis Lloyd's mirror interferometer,which has been realized through innovation to Lloyd’s mirror interferometer,has been developed.In addition,the concept of the patterning of the two-dimensional orthogonal pattern structure at a single exposure has been extended to the non-orthogonal two-axis Lloyd’s mirror interferometer.Furthermore,the optical setup for the non-orthogonal two-axis Lloyd's mirror interferometer has been optimized for the fabrication of a large-area scale grating.In this review article,principles of generating interference fringe fields for the fabrication of a scale grating based on the interference lithography are reviewed,while focusing on the fabrication of a two-dimensional scale grating for planar/surface encoders.Verification of the pitch of the fabricated pattern structures,whose accuracy strongly affects the performance of planar/surface encoders,is also an important task to be addressed.In this paper,major methods for the evaluation of a grating pitch are also reviewed.展开更多
基金supported by the U.S.Department of Energy under Award DE-SC-0001691support from the ORAU Ralph E.Powe Junior Faculty Enhancement Award and from the National Science Foundation under the grants DMS-1056821 and DMS-0915013.
文摘In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs,are in big demand due to their optimality properties important for many applications.The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings.This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems.Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems,and O(k)complexity estimation is provided for a problem with k generators.
基金supported by the Japan Society for the Promotion of Science(JSPS).The author would like to thank Professor Wei Gao(Tohoku University,Japan)for the fruitful discussions and valuable comments on the topic of this review article.
文摘Laser interference lithography is an attractive method for the fabrication of a large-area two-dimensional planar scale grating,which can be employed as a scale for multi-axis optical encoders or a diffractive optical element in many types of optical sensors.Especially,optical configurations such as Lloyd's mirror interferometer based on the division of wavefront method can generate interference fringe fields for the patterning of grating pattern structures at a single exposure in a stable manner.For the fabrication of a two-dimensional scale grating to be used in a planar/surface encoder,an orthogonal two-axis Lloyd's mirror interferometer,which has been realized through innovation to Lloyd’s mirror interferometer,has been developed.In addition,the concept of the patterning of the two-dimensional orthogonal pattern structure at a single exposure has been extended to the non-orthogonal two-axis Lloyd’s mirror interferometer.Furthermore,the optical setup for the non-orthogonal two-axis Lloyd's mirror interferometer has been optimized for the fabrication of a large-area scale grating.In this review article,principles of generating interference fringe fields for the fabrication of a scale grating based on the interference lithography are reviewed,while focusing on the fabrication of a two-dimensional scale grating for planar/surface encoders.Verification of the pitch of the fabricated pattern structures,whose accuracy strongly affects the performance of planar/surface encoders,is also an important task to be addressed.In this paper,major methods for the evaluation of a grating pitch are also reviewed.
