Method and apparatus for estimating model parameters of and controlling a lithographic apparatus by measuring a substrate property and using a polynomial model

An apparatus for estimating model parameters of a lithographic apparatus used to perform a lithographic process across a substrate, the apparatus comprising a processor configured to:

• use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
• receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
• and
• calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate.

An apparatus for controlling lithographic processing by a lithographic apparatus used to perform a lithographic process across a substrate, the apparatus comprising a processor configured to:

• use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
• receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
• calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate;
• and
• control lithographic processing by the lithographic apparatus using the estimated model parameters.

A non-transitory computer readable medium having instructions stored thereon for estimating model parameters of a lithographic apparatus used to perform a lithographic process across a substrate, the instructions when executed by one or more processors, cause the one or more processors to:

• use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
• receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
• and
• calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate.

A non-transitory computer readable medium having instructions stored thereon for controlling lithographic processing by a lithographic apparatus used to perform a lithographic process across a substrate, the instructions when executed by one or more processors, cause the one or more processors to:

• use predetermined substrate measurement locations to generate discrete orthogonal polynomials;
• receive substrate property measurements arising from the lithographic process measured at locations of the substrate corresponding to the predetermined measurement locations;
• calculate estimated model parameters of the lithographic apparatus from the substrate property measurements by using the generated discrete orthogonal polynomials as a basis function to model the substrate property across the substrate;
• and
• control lithographic processing by the lithographic apparatus using the estimated model parameters.