Hessian of Image Log Slope for Source Mask Optimization
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Solution Overview
Problem
The lithographic process for optimizing mask and source design is time-consuming and laborious due to the need for frequent fine-tuning and repetition of source-mask optimization processes, especially in ensuring sufficient Image Log Slope (ILS) across critical and non-critical areas of the mask design layout.
Innovation Solution
Determining the gradient and Hessian of the Image Log Slope (ILS) at selected points on the mask design layout to optimize the mask pattern and illumination source, allowing for the maximization of ILS without requiring continuous fine-tuning of the source-mask optimization process, and incorporating sub-resolution assist features to improve imaging.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If conventional source-mask optimization process is used to ensure sufficient Image Log Slope across critical and non-critical areas, then imaging quality is improved, but the optimization process becomes time-consuming and laborious due to frequent fine-tuning and repetition
Solution Approach 1:
The patent applies preliminary action by pre-calculating and storing the Hessian matrix of the Image Log Slope function before the optimization process begins. This pre-computed information is then reused during optimization iterations, eliminating the need for repeated full gradient calculations and reducing the number of fine-tuning cycles required to achieve sufficient ILS across critical and non-critical areas.
Solution Approach 2:
The patent changes the parameter representation by working with the Hessian matrix of the ILS function rather than directly optimizing the mask and source parameters. This transformation allows for more efficient convergence by capturing second-order information about the optimization landscape, thereby reducing iterative refinement time while maintaining imaging quality.
2Manufacturing precision
If continuous fine-tuning of source-mask optimization is performed to maximize Image Log Slope, then imaging quality is improved, but the process requires repetitive optimization cycles
Solution Approach 1:
The patent implements feedback by using the pre-computed Hessian matrix to guide the optimization direction more accurately. This second-order information provides better feedback about the curvature of the ILS function, allowing the optimization algorithm to take more informed steps and converge faster, thereby reducing repetitive cycling while maximizing Image Log Slope.
Solution Approach 2:
The patent applies dynamics by adapting the optimization strategy based on the pre-calculated Hessian information. Instead of using a static first-order gradient approach, the method dynamically adjusts optimization steps using second-order derivative information, enabling faster convergence and reducing the number of repetitive optimization cycles needed to achieve maximum ILS.
3Manufacturing precision
If design rules are optimized for critical areas first, then critical dimension control is improved, but non-critical areas require separate re-optimization
Solution Approach 1:
The patent applies universality by using a single Hessian matrix calculation that captures the interactions between all mask and source parameters across both critical and non-critical areas. This unified second-order information allows simultaneous optimization of both critical dimension control and non-critical area performance within one optimization framework, eliminating the need for separate re-optimization steps.
Solution Approach 2:
The patent merges the optimization of critical and non-critical areas into a single unified process by using the pre-computed Hessian matrix that encompasses all evaluation points. This combination allows the optimization algorithm to simultaneously satisfy both critical dimension requirements and non-critical area performance, reducing process complexity while maintaining comprehensive control.
Data Source
AI summary
The Hessian (second derivative) of the image log slope (ILS) can be quickly and accurately calculated without the need to use approximate methods from the gradient of the ILS with respect to mask transmission and source intensity. The Hessian has been traditionally calculated using a finite-difference approach. Calculating the Hessian through a finite-difference approach is slow and is an approximate method. The gradient of the ILS improves the speed of calculation of the Hessian, and thus accelerated SMO operation is realized. The results of ILS evaluation can be used in design for manufacturing (DFM) to suggest changes in the design rules to improve imaging. For a fixed illumination, this information can help remove forbidden pitches and help select design rules for 1-D and 2-D patterns on a mask design layout.