A method for measuring a parameter of interest from a target and associated apparatuses. The method includes obtaining measurement acquisition data relating to measurement of the target and finite-size effect correction data and/or a trained model operable to correct for at least finite-size effects in the measurement acquisition data. At least finite-size effects in the measurement acquisition data is corrected for using the finite-size effect correction data and/or the trained model to obtain corrected measurement data and/or obtain a parameter of interest; and where the correcting does not directly determine the parameter of interest, determining the parameter of interest from the corrected measurement data.

A method for determining a spatially varying process offset for a lithographic process, the spatially varying process offset (MTD) varying over a substrate subject to the lithographic process to form one or more structures thereon. The method includes obtaining a trained model (MOD), having been trained to predict first metrology data based on second metrology data, wherein the first metrology data (OV) is spatially varying metrology data which relates to a first type of measurement of the one or more structures being a measure of yield and the second metrology data (PB) is spatially varying metrology data which relates to a second type of measurement of the one or more structures and correlates with the first metrology data; and using the model to obtain the spatially varying process offset (MTD).

A metrology system may include an optical metrology tool configured to produce an optical metrology output for one or more features on a processed substrate, and a metrology machine learning model that has been trained using a training set of (i) profiles, critical dimensions, and/or contours for a plurality of features, and (ii) optical metrology outputs for the plurality of features. The metrology machine learning model may be configured to: receive the optical metrology output from the optical metrology tool; and output the profile, critical dimension, and/or contour of the one or more features on the processed substrate.

Disclosed is a method of determining a performance parameter or a parameter derived therefrom, the performance parameter being associated with a performance of a lithographic process for forming one or more structures on a substrate subject to the lithographic process. The method comprises obtaining a probability description distribution comprising a plurality of probability descriptions of the performance parameter, each probability description corresponding to a different position on the substrate and decomposing each probability description into a plurality of component probability descriptions to obtain a plurality of component probability description distributions. A component across-substrate-area model is determined for each of said plurality of component probability descriptions, which models its respective component probability description across a substrate area; and a value for said performance parameter or parameter derived therefrom is determined based on the component across-substrate-area models.

A modular autoencoder model is described. The modular autoencoder model comprises input models configured to process one or more inputs to a first level of dimensionality suitable for combination with other inputs; a common model configured to: reduce a dimensionality of combined processed inputs to generate low dimensional data in a latent space; and expand the low dimensional data in the latent space into one or more expanded versions of the one or more inputs suitable for generating one or more different outputs; output models configured to use the one or more expanded versions of the one or more inputs to generate the one or more different outputs, the one or more different outputs being approximations of the one or more inputs; and a prediction model configured to estimate one or more parameters based on the low dimensional data in the latent space.

A modular autoencoder model is described. The modular autoencoder model comprises input models configured to process one or more inputs to a first level of dimensionality suitable for combination with other inputs; a common model configured to: reduce a dimensionality of combined processed inputs to generate low dimensional data in a latent space; and expand the low dimensional data in the latent space into one or more expanded versions of the one or more inputs suitable for generating one or more different outputs; output models configured to use the one or more expanded versions of the one or more inputs to generate the one or more different outputs, the one or more different outputs being approximations of the one or more inputs; and a prediction model configured to estimate one or more parameters based on the low dimensional data in the latent space.

A method including obtaining a measurement result from a target on a substrate, by using a substrate measurement recipe; determining, by a hardware computer system, a parameter from the measurement result, wherein the parameter characterizes dependence of the measurement result on an optical path length of the target for incident radiation used in the substrate measurement recipe and the determining the parameter includes determining dependence of the measurement result on a relative change of wavelength of the incident radiation; and if the parameter is not within a specified range, adjusting the substrate measurement recipe.

A method for calibrating a resist model. The method includes: generating a modeled resist contour of a resist structure based on a simulated aerial image of the resist structure and parameters of the resist model, and predicting a metrology contour of the resist structure from the modeled resist contour based on information of an actual resist structure obtained by a metrology device. The method includes adjusting one or more of the parameters of the resist model based on a comparison of the predicted metrology contour and an actual metrology contour of the actual resist structure obtained by the metrology device.

A method for determining a value representative of a state transition temperature of a resist structure, formed of a resist material and having predetermined dimensions, on an underlayer material includes: receiving data earlier obtained, the data representing a correlation between a second value for a measure representative of a spatial feature of at least one resist structure of each of a plurality of entities after applying a heat treatment, and a temperature at which the heat treatment is applied, each entity comprising the at least one resist structure, formed of the resist material and having the predetermined dimensions before the heat treatment, on the underlayer material, and wherein the measure has a first value before the heat treatment, and determining, from the correlation, the value representative of the state transition temperature when the heat treatment would be performed at such temperature, the second value differs by a predetermined amount from the first value.

A method for generating a retargeted pattern for a target pattern to be printed on a substrate. The method includes obtaining (i) the target pattern comprising at least one feature, the at least one feature having geometry including a first dimension and a second dimension, and (ii) a plurality of biasing rules defined as a function of the first dimension, the second dimension, and a property associated with features of the target pattern within a measurement region; determining values of the property at a plurality of locations on the at least one feature of the target pattern, each location surrounded by the measurement region; selecting, from the plurality of biasing rules based on the values of the property, a sub-set of biases; and generating the retargeted pattern by applying the selected sub-set of biases to the at least one feature of the target pattern.