Determining the gradient and Hessian of the image log slope for design rule optimization for accelerating source mask optimization (SMO)

A non-transitory computer-readable medium containing instructions for a computer to improve a design of mask patterns to be imaged using a lithographic process by illuminating the mask patterns with an illumination source in a lithographic apparatus, the design improvement process comprising:

• receiving an initial design layout of the mask patterns, wherein the mask patterns include a known critical dimension for which a calculated value of an image log slope (ILS) at one or more evaluation points on an image plane is below a predefined threshold value;
• calculating a second order partial derivative of the ILS at the one or more evaluation points with respect to a first variable and a second variable, the first variable being associated with mask transmission, and, the second variable being associated with an intensity at a point on the illumination source;
• and
• using the calculated second order partial derivative to determine a correction amount by which to adjust the critical dimension to improve the value of the ILS at the one or more evaluation points.

A computer-implemented method for accelerating a source mask optimization (SMO) process by adjusting mask patterns to be imaged using a lithographic process and an illumination source in a lithographic apparatus, the method comprising:

• receiving an initial design layout of the mask patterns, wherein the mask patterns include a known critical dimension for which a calculated value of an image log slope (ILS) at one or more evaluation points on an image plane is below a predefined threshold value;
• calculating a gradient of the ILS at the one or more evaluation points with respect to any one of a first variable and a second variable, the first variable being associated with mask transmission, and, the second variable being associated with an intensity at a point on the illumination source;
• calculating a second order partial derivative of the ILS at the one or more evaluation points using the calculated gradient of the ILS with respect to the remaining variable that is not used in calculating the gradient of the ILS;
• and
• using the calculated second order partial derivative to improve the value of the ILS at the one or more evaluation points by adjusting one or both of the mask patterns and the illumination source, thereby accelerating the SMO process,
• wherein the steps of the method are performed by the computer.