Rational approximation and continued-fraction approximation approaches for computation efficiency of diffraction signals

1. A method for improving computation efficiency for diffraction signals for optical metrology, comprising:

simulating a set of diffraction orders for a structure, the structure one of a plurality of repeating structures on a wafer;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix rational approximation to the set of diffraction efficiencies to obtain a rationally approximated set of diffraction efficiencies; and

providing a simulated spectrum for the structure based on the rationally approximated set of diffraction efficiencies;

generating a sample spectrum from an optical metrology system; and

comparing the simulated spectrum to the sample spectrum.

2. The method of claim 1 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization.

3. The method of claim 1 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using a rigorous coupled wave approach.

4. The method of claim 1 , further comprising:

performing a regress analysis on the rationally approximated set of diffraction efficiencies prior to providing the simulated spectrum.

5. The method of claim 4 , wherein performing the regress analysis comprises using an analytic Jacobian-based method.

6. The method of claim 1 , wherein simulating the set of diffraction orders comprises simulating the set of diffraction orders for a structure selected from the group consisting of a two-dimensional structure, a three-dimensional structure, and a structure having both a three-dimensional component and a two-dimensional component.

7. A method for improving computation efficiency for diffraction signals for optical metrology, comprising:

simulating a set of diffraction orders for a structure, the structure one of a plurality of repeating structures on a wafer;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix continued-fraction approximation to the set of diffraction efficiencies to obtain a continued-fraction approximated set of diffraction efficiencies; and

providing a simulated spectrum for the structure based on the continued-fraction approximated set of diffraction efficiencies;

generating a sample spectrum from an optical metrology system; and

comparing the simulated spectrum to the sample spectrum.

8. The method of claim 7 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization.

9. The method of claim 7 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using a rigorous coupled wave approach.

10. The method of claim 7 , further comprising:

performing a regress analysis on the continued-fraction approximated set of diffraction efficiencies prior to providing the simulated spectrum.

11. The method of claim 10 , wherein performing the regress analysis comprises using an analytic Jacobian-based method.

12. The method of claim 7 , wherein simulating the set of diffraction orders comprises simulating the set of diffraction orders for a structure selected from the group consisting of a two-dimensional structure, a three-dimensional structure, and a structure having both a three-dimensional component and a two-dimensional component.

13. A non-transitory computer readable medium containing instructions stored therein for causing a computer processor to perform a method for improving computation efficiency for diffraction signals in optical metrology, the method comprising:

simulating a set of diffraction orders for a structure;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix rational approximation to the set of diffraction efficiencies to obtain a rationally approximated set of diffraction efficiencies; and

providing a simulated spectrum based on the rationally approximated set of diffraction efficiencies.

14. The non-transitory computer readable medium as in claim 13 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization or using a rigorous coupled wave approach.

15. The non-transitory computer readable medium as in claim 13 , having instructions stored therein for causing the computer processor to perform the method further comprising:

performing a regress analysis on the rationally approximated set of diffraction efficiencies prior to providing the simulated spectrum, wherein performing the regress analysis comprises using an analytic Jacobian-based method.

16. The non-transitory computer readable medium as in claim 13 , having instructions stored therein for causing the computer processor to perform the method further comprising:

comparing the simulated spectrum to a sample spectrum.

17. The non-transitory computer readable medium as in claim 13 , wherein simulating the set of diffraction orders comprises simulating the set of diffraction orders for a structure selected from the group consisting of a two-dimensional structure, a three-dimensional structure, and a structure having both a three-dimensional component and a two-dimensional component.

18. A non-transitory computer readable medium containing instructions stored therein for causing a computer processor to perform a method for improving computation efficiency for diffraction signals in optical metrology, the method comprising:

simulating a set of diffraction orders for a structure;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix continued-fraction approximation to the set of diffraction efficiencies to obtain a continued-fraction approximated set of diffraction efficiencies; and

providing a simulated spectrum based on the continued-fraction approximated set of diffraction efficiencies.

19. The non-transitory computer readable medium as in claim 18 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization or using a rigorous coupled wave approach.

20. The non-transitory computer readable medium as in claim 18 , having instructions stored therein for causing the computer processor to perform the method further comprising:

performing a regress analysis on the continued-fraction approximated set of diffraction efficiencies prior to providing the simulated spectrum, wherein performing the regress analysis comprises using an analytic Jacobian-based method.

21. The non-transitory computer readable medium as in claim 18 , having instructions stored therein for causing the computer processor to perform the method further comprising:

comparing the simulated spectrum to a sample spectrum.

22. The non-transitory computer readable medium as in claim 18 , wherein simulating the set of diffraction orders comprises simulating the set of diffraction orders for a structure selected from the group consisting of a two-dimensional structure, a three-dimensional structure, and a structure having both a three-dimensional component and a two-dimensional component.

23. A system for improving computation efficiency for diffraction signals in optical metrology, comprising:

a processor;

a main memory;

a secondary memory having instructions stored thereon which cause the system to perform a method, comprising:

simulating a set of diffraction orders for a structure;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix rational approximation to the set of diffraction efficiencies to obtain a rationally approximated set of diffraction efficiencies; and

providing a simulated spectrum based on the rationally approximated set of diffraction efficiencies; and

a bus which allows the processor, the main memory and the secondary memory to communicate with one another.

24. The system as in claim 23 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization.

25. The system as in claim 23 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using a rigorous coupled wave approach.

26. A system for improving computation efficiency for diffraction signals in optical metrology, comprising:

a processor;

a main memory;

a secondary memory having instructions stored thereon which cause the system to perform a method, comprising:

simulating a set of diffraction orders for a structure;

determining a set of diffraction efficiencies for the set of diffraction orders;

applying a matrix continued-fraction approximation to the set of diffraction efficiencies to obtain a continued-fraction approximated set of diffraction efficiencies; and

providing a simulated spectrum based on the continued-fraction approximated set of diffraction efficiencies; and

a bus which allows the processor, the main memory and the secondary memory to communicate with one another.

27. The system as in claim 26 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using modified Legendre polynomial high-order finite element discretization.

28. The system as in claim 26 , wherein determining the set of diffraction efficiencies for the set of diffraction orders comprises using a rigorous coupled wave approach.