A new method for aspherical surface fitting with large-volume datasets

In the framework of form characterization of aspherical surfaces, European National Metrology Institutes (NMIs) have been developing ultra-high precision machines having the ability to measure aspherical lenses with an uncertainty of few tens of nanometers. The fitting of the acquired aspherical datasets onto their corresponding theoretical model should be achieved at the same level of precision. In this article, three fitting algorithms are investigated: the Limited memory-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), the Levenberg–Marquardt (LM) and one variant of the Iterative Closest Point (ICP). They are assessed based on their capacities to converge relatively fast to achieve a nanometric level of accuracy, to manage a large volume of data and to be robust to the position of the data with respect to the model. Nev-ertheless, the algorithms are first evaluated on simulated datasets and their performances are studied. The comparison of these algorithms is extended on measured datasets of an aspherical lens. The results validate the newly used method for the fitting of aspherical surfaces and reveal that it is well adapted, faster and less complex than the LM or ICP methods.EMR

Review of the mathematical foundations of data fusion techniques in surface metrology

The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed

Reduced basis method for source mask optimization

Image modeling and simulation are critical to extending the limits of leading edge lithography technologies used for IC making. Simultaneous source mask optimization (SMO) has become an important objective in the field of computational lithography. SMO is considered essential to extending immersion lithography beyond the 45nm node. However, SMO is computationally extremely challenging and time-consuming. The key challenges are due to run time vs. accuracy tradeoffs of the imaging models used for the computational lithography. We present a new technique to be incorporated in the SMO flow. This new approach is based on the reduced basis method (RBM) applied to the simulation of light transmission through the lithography masks. It provides a rigorous approximation to the exact lithographical problem, based on fully vectorial Maxwell's equations. Using the reduced basis method, the optimization process is divided into an offline and an online steps. In the offline step, a RBM model with variable geometrical parameters is built self-adaptively and using a Finite Element (FEM) based solver. In the online step, the RBM model can be solved very fast for arbitrary illumination and geometrical parameters, such as dimensions of OPC features, line widths, etc. This approach dramatically reduces computational costs of the optimization procedure while providing accuracy superior to the approaches involving simplified mask models. RBM furthermore provides rigorous error estimators, which assure the quality and reliability of the reduced basis solutions. We apply the reduced basis method to a 3D SMO example. We quantify performance, computational costs and accuracy of our method.Comment: BACUS Photomask Technology 201

Measurement cost of metric-aware variational quantum algorithms

Variational quantum algorithms are promising tools for near-term quantum computers as their shallow circuits are robust to experimental imperfections. Their practical applicability, however, strongly depends on how many times their circuits need to be executed for sufficiently reducing shot-noise. We consider metric-aware quantum algorithms: variational algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the quantum Fisher information matrix as a metric tensor to correct the gradient vector for the co-dependence of the circuit parameters. We rigorously characterise and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.Comment: 17 pages, 3 figure