Method for determining the registration of a structure on a photomask and apparatus to perform the method

1. A computer-implemented method comprising:

simulating, using a processor, a second aerial image from pattern specifications of a mask, including determining said second aerial image as a matrix composed of pixels having intensity values, taking as a basis the pattern specifications of the mask in the form of a transmission function that has, at least in part, a higher resolution than a first aerial image of the mask that was captured by a position measuring device, the first aerial image having a feature, the second aerial images also having the feature; and

determining, using the processor, a registration error of the feature on the mask based on a comparison of the simulated second aerial image and the first aerial image, the registration error indicating a deviation of a nominal position of the feature from its actual position on the mask.

2. The method as in claim 1 , wherein the pattern specifications of the mask are in the form of a continuous transmission function, a discretization of the spectrum of the mask being performed without further approximation.

3. The method as in claim 2 , wherein said continuous transmission function is decomposed into a sum of functions that describe the basic building blocks of the pattern specifications.

4. The method as in claim 3 , wherein said building blocks are embodied as triangles.

5. The method as in claim 1 , wherein the pattern specifications of the mask are transformed into a matrix.

6. The method as in claim 5 , wherein the resolution of said matrix varies.

7. The method of claim 1 in which the first aerial image of the mask is a first matrix composed of pixels having intensity values, the second aerial image of the mask is a second matrix composed of pixels having intensity values, and each pixel in the second matrix represents a smaller dimension on the mask than that of a corresponding pixel in the first matrix.

8. The method of claim 7 in which each pixel in the first matrix represents a dimension on the mask that is at least 10 times larger than that of a corresponding pixel in the second matrix.

9. The method of claim 3 in which each building block has a constant complex transmission.

10. The method of claim 9 in which the mask has patterns formed by a patterned layer of opaque material on a clear substrate, in which regions of the mask coated with the opaque material are decomposed into building blocks of a first complex transmission, and regions of the mask that is free of the opaque material are decomposed into building blocks of a second complex transmission.

11. A position measuring device for determining a registration error of a feature on a mask, the position measuring device comprising:

a control unit that is configured to determine a registration error of a feature on a mask, the registration error indicating a deviation of a nominal position of the feature from its actual position on the mask,

wherein the control unit is configured to simulate a second aerial image from pattern specifications of a mask, including determining said second aerial image as a matrix composed of pixels having intensity values, taking as a basis the pattern specifications of the mask in the form of a transmission function that has, at least in part, a higher resolution than a first aerial image of the mask that was captured by a position measuring device, the first aerial image having a feature, the second aerial images also having the feature, and

wherein the control unit is configured to determine a registration error of the feature on the mask based on a comparison of the simulated second aerial image and the first aerial image.

12. The position measuring device of claim 11 in which the pattern specifications of the mask are in the form of a continuous transmission function, a discretization of the spectrum of the mask being performed without further approximation.

13. The position measuring device of claim 12 in which the continuous transmission function is decomposed into a sum of functions that describe the basic building blocks of the pattern specifications.

14. The position measuring device of claim 13 in which the building blocks are embodied as triangles.

15. The position measuring device of claim 13 in which each building block has a constant complex transmission.

16. The position measuring device of claim 15 in which the mask has patterns formed by a patterned layer of opaque material on a clear substrate, in which regions of the mask coated with the opaque material are decomposed into building blocks of a first complex transmission, and regions of the mask that is free of the opaque material are decomposed into building blocks of a second complex transmission.

17. The position measuring device of claim 11 in which the pattern specifications of the mask are transformed into a matrix.

18. The position measuring device of claim 17 in which the resolution of the matrix varies.

19. The position measuring device of claim 11 in which in which the first aerial image of the mask is a first matrix composed of pixels having intensity values, the second aerial image of the mask is a second matrix composed of pixels having intensity values, and each pixel in the second matrix represents a smaller dimension on the mask than that of a corresponding pixel in the first matrix.

20. The position measuring device of claim 19 in which each pixel in the first matrix represents a dimension on the mask that is at least 10 times larger than that of a corresponding pixel in the second matrix.