Inspection method and apparatus, and corresponding lithographic apparatus

A lithographic apparatus comprising:

• an illumination system configured to condition a radiation beam;
• a support configured to support a patterning device, the patterning device being capable of imparting the radiation beam with a pattern in its cross-section to form a patterned radiation beam;
• a substrate table configured to hold a substrate;
• a projection system configured to project the patterned radiation beam onto a target portion of the substrate;
• and
• an inspection tool comprising:
• a scatterometer having an optical system with a high numerical aperture of at least 0.9, the scatterometer configured to:
• illuminate a plurality of structures formed on the substrate;
• acquire diffraction pupil images of the plurality of structures formed on the substrate during a lithographic process, a process variable of the lithographic process having been varied between formation of the structures, the variation of the process variable resulting in a variation in the formed structures and consequently the diffraction pupil images, wherein the acquiring of the diffraction pupil image is performed using a detector arranged at a back-projected pupil plane or at a reimage back-projected pupil plane of the scatterometer;
• and
• determine at least one discriminant function for the diffraction pupil images, the discriminant function being able to classify the pupil images in terms of the process variable,
• wherein the discriminant function includes at least one of logistic regression and support vector machines.

A non-transitory computer program product comprising instructions for causing a programmable data processing apparatus to perform as the data processing apparatus performing a method comprising:

• illuminating a plurality of structures using a scatterometer optical system with a high numerical aperture of at least 0.9;
• acquiring diffraction pupil images of the plurality of structures formed on a substrate during a lithographic process, a process variable of the lithographic process having been varied between formation of the structures, the variation of the process variable resulting in a variation in the formed structures and consequently the diffraction pupil images, wherein the acquiring of the diffraction pupil image is performed using a detector arranged at a back-projected pupil plane or at a reimage back-projected pupil plane of the scatterometer;
• and
• determining at least one discriminant function for the diffraction pupil images, the discriminant function being able to classify the pupil images in terms of the process variable,
• wherein the determining discriminant function includes at least one of logistic regression and support vector machines.