Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures

Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products

Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures

Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products

Training procedure for scanning electron microscope 3D surface reconstruction using unsupervised domain adaptation with simulated data

Accurate metrology techniques for semiconductor devices are indispensable for controlling the manufacturing process. For instance, the dimensions of a transistor’s current channel (fin) are an important indicator of the device’s performance regarding switching voltages and parasitic capacities. We expand upon traditional 2D analysis by utilizing computer vision techniques for full-surface reconstruction. We propose a data-driven approach that predicts the dimensions, height and width (CD) values, of fin-like structures. During operation, the method solely requires experimental images from a scanning electron microscope of the patterns concerned. We introduce an unsupervised domain adaptation step to overcome the domain gap between experimental and simulated data. Our model is further fine-tuned with a height measurement from a second scatterometry sensor and optimized through a tailored training scheme for optimal performance. The proposed method results in accurate depth predictions, namely 100% accurate interwafer classification with an root-mean-squared error of 0.67 nm. The R2 of the intrawafer performance on height is between 0.59 and 0.70. Qualitative results also indicate that detailed surface features, such as corners, are accurately predicted. Our study shows that accurate z-metrology techniques can be viable for high-volume manufacturing

Efficient Computation of Three-Dimensional Flow in Helically Corrugated Hoses Including Swirl

In this article we propose an efficient method to compute the friction factor of helically corrugated hoses carrying flow at high Reynolds numbers. A comparison between computations of several turbulence models is made with experimental results for corrugation sizes that fall outside the range of validity of the Moody diagram. To do this efficiently we implement quasi-periodicity. Using the appropriate boundary conditions and matching body force, we only need to simulate a single period of the corrugation to find the friction factor for fully developed flow. A second technique is introduced by the construction of an appropriately twisted wedge, which allows us to furthermore reduce the problem by a further dimension while accounting for the Beltrami symmetry that is present in the full three-dimensional problem. We make a detailed analysis of the accuracy and time-saving that this novelty introduces. We show that the swirl inside the flow, which is introduced by the helical boundary, has a positive effect on the friction factor. Furthermore, we give a prediction for which corrugation angles the assumption of axisymmetry is no longer valid. It then has to make place for Beltrami-symmetry if accurate results are required

Modeling and optimization of algae growth

The wastewater from greenhouses has a high amount of mineral contamination\ud and an environmentally-friendly method of removal is to use algae\ud to clean this runoff water. The algae consume the minerals as part of their\ud growth process. In addition to cleaning the water, the created algal bio-mass\ud has a variety of applications including production of bio-diesel, animal feed,\ud products for pharmaceutical and cosmetic purposes, or it can even be used as\ud a source of heating or electricity .\ud The aim of this paper is to develop a model of algae production and use\ud this model to investigate how best to optimize algae farms to satisfy the dual\ud goals of maximizing growth and removing mineral contaminants.\ud With this aim in mind the paper is split into five main sections. In the\ud first a review of the biological literature is undertaken with the aim of determining\ud what factors effect the growth of algae. The second section contains\ud a review of exciting mathematical models from the literature, and for\ud each model a steady-state analysis is performed. Moreover, for each model\ud the strengths and weaknesses are discussed in detail. In the third section,a new two-stage model for algae production is proposed, careful estimation\ud of parameters is undertaken and numerical solutions are presented. In the\ud next section, a new one-dimensional spatial-temporal model is presented,\ud numerically solved and optimization strategies are discussed. Finally, these\ud elements are brought together and recommendations of how to continue are\ud drawn

Apparatus and method for determining three dimensional data based on an image of a patterned substrate

Described herein are system, method, and apparatus for determining three-dimensional (3D) information of a structure of a patterned substrate. The 3D information can be determined using one or more model configured to generate 3D information (e.g., depth information) using only a single image of a patterned substrate. In a method, the model is trained by obtaining a pair of stereo images of a structure of a patterned substrate. The model generates, using a first image of the pair of stereo images as input, disparity data between the first image and a second image, the disparity data being indicative of depth information associated with the first image. The disparity data is combined with the second image to generate a reconstructed image corresponding to the first image. Further, one or more model parameters are adjusted based on the disparity data, the reconstructed image, and the first image

Apparatus and method for determining three dimensional data based on an image of a patterned substrate

Described herein are system, method, and apparatus for determining three-dimensional (3D) information of a structure of a patterned substrate. The 3D information can be determined using one or more model configured to generate 3D information (e.g., depth information) using only a single image of a patterned substrate. In a method, the model is trained by obtaining a pair of stereo images of a structure of a patterned substrate. The model generates, using a first image of the pair of stereo images as input, disparity data between the first image and a second image, the disparity data being indicative of depth information associated with the first image. The disparity data is combined with the second image to generate a reconstructed image corresponding to the first image. Further, one or more model parameters are adjusted based on the disparity data, the reconstructed image, and the first image