Method, program product and apparatus for optimizing illumination for full-chip layer

1. A method of optimizing illumination for a full-chip layer, comprising the steps of:

determining a pitch frequency of the full-chip layer, the pitch frequency indicating how often a pitch occurs in the full-chip layer so as to generate a pitch frequency histogram of the full-chip layer;

equating the pitch frequency histogram to be a first eigenfunction of the sum of coherent systems representation of a transformation cross coefficient; and

solving an integral equation for the first eigenfunction of the transformation cross coefficient so as to define the optimal illumination for the full-chip layer.

2. The method according to claim 1 , wherein all pitches are equally weighted in the pitch frequency histogram.

3. The method according to claim 1 , wherein the integral equation for the first eigenfunction of the transformation cross coefficient is solved such that the first eigenfunction becomes substantially identical to the pitch frequency.

4. The method according to claim 3 , wherein a difference between the pitch frequency and the first eigenfunction is minimized by using a merit function to solve the integral equation for the first eigenfunction of the transformation cross coefficient.

5. The method according to claim 1 , wherein the merit function is essentially the convolution of the first eigenfunction with the pitch frequency.

6. The method according to claim 1 , wherein the integral equation for the first eigenfunction of the transformation cross coefficient is solved such that the first eigenfunction has a value of one for every non-zero value in the pitch frequency.

7. The method according to claim 1 , wherein the pitch frequency is obtained by determining the separation between adjacent features on the full-chip layer that are within a coherence radius.

8. The method according to claim 1 , wherein the pitch frequency is obtained by representing adjacent features on the full-chip layer with Dirac delta functions and measuring separation between the Dirac delta functions.

9. The method according to claim 1 , wherein the pitch frequency for a random hole pattern or for a periodic hole pattern is obtained as a function of hole separation and as a function of angular separation.

10. The method according to claim 1 , further comprising:

determining whether pitch frequencies fall into a negative region of the first eigenfunction by using the optimized illumination;

separating the pitch frequencies which fall into the negative region into another layer to be imaged with a second exposure; and

optimizing illumination for said another layer.

11. A computer program product having a computer readable storage medium bearing a computer program for optimizing illumination for a full-chip layer, the computer program, which executed, causing a computer to perform the steps of:

determining a pitch frequency of the full-chip layer, the pitch frequency indicating how often a pitch occurs in the full-chip layer so as to generate a pitch frequency histogram of the full-chip layer;

equating the pitch frequency histogram to be a first eigenfunction of the sum of coherent systems representation of a transformation cross coefficient; and

solving an integral equation for the first eigenfunction of the transformation cross coefficient so as to define the optimal illumination for the full-chip layer.

12. The computer program product according to claim 11 , wherein all pitches are equally weighted in the pitch frequency histogram.

13. The computer program product according to claim 11 , wherein the integral equation for the first eigenfunction of the transformation cross coefficient is solved such that the first eigenfunction becomes substantially identical to the pitch frequency.

14. The computer program product according to claim 13 , wherein a difference between the pitch frequency and the first eigenfunction is minimized by using a merit function to solve the integral equation for the first eigenfunction of the transformation cross coefficient.

15. The computer program product according to claim 11 , wherein the merit function is essentially the convolution of the first eigenfunction with the pitch frequency.

16. The computer program product according to claim 11 , wherein the integral equation for the first eigenfunction of the transformation cross coefficient is solved such that the first eigenfunction has a value of one for every non-zero value in the pitch frequency.

17. The computer program product according to claim 11 , wherein the pitch frequency is obtained by determining the separation between adjacent features on the full-chip layer that are within a coherence radius.

18. The computer program product according to claim 11 , wherein the pitch frequency is obtained by representing adjacent features on the full-chip layer with Dirac delta functions and measuring separation between the Dirac delta functions.

19. The computer program product according to claim 11 , wherein the pitch frequency for a random hole pattern or for a periodic hole pattern is obtained as a function of hole separation and as a function of angular separation.

20. The computer program product according to claim 11 , further comprising:

determining whether pitch frequencies fall into a negative region of the first eigenfunction by using the optimized illumination;

separating the pitch frequencies which fall into the negative region into another layer to be imaged with a second exposure; and

optimizing illumination for said another layer.

21. An apparatus for optimizing illumination for a full-chip layer, comprising:

a first unit configured for determining a pitch frequency of the full-chip layer, the pitch frequency indicating how often a pitch occurs in the full-chip layer so as to generate a pitch frequency histogram of the full-chip layer;

a second unit configured for equating the pitch frequency histogram to be a first eigenfunction of the sum of coherent systems representation of a transformation cross coefficient; and

a third unit configured for solving an integral equation for the first eigenfunction of the transformation cross coefficient so as to define the optimal illumination for the full-chip layer.

22. The apparatus according to claim 21 , wherein the first unit weights all pitches equally in the pitch frequency histogram.

23. The apparatus according to claim 21 , wherein the third unit solves the integral equation for the first eigenfunction of the transformation cross coefficient such that the first eigenfunction becomes substantially identical to the pitch frequency.

24. The apparatus according to claim 23 , wherein the third unit minimizes a difference between the pitch frequency and the first eigenfunction by using a merit function to solve the integral equation for the first eigenfunction of the transformation cross coefficient.

25. The apparatus according to claim 21 , wherein the merit function is essentially the convolution of the first eigenfunction with the pitch frequency.

26. The apparatus according to claim 21 , wherein the third unit solves the integral equation for the first eigenfunction of the transformation cross coefficient such that the first eigenfunction has a value of one for every non-zero value in the pitch frequency.

27. The apparatus according to claim 21 , wherein the first unit obtains the pitch frequency by determining the separation between adjacent features on the full-chip layer that are within a coherence radius.

28. The apparatus according to claim 21 , wherein the first unit obtains the pitch frequency by representing adjacent features on the full-chip layer with Dirac delta functions and measuring separation between the Dirac delta functions.

29. The apparatus according to claim 21 , wherein the first unit obtains the pitch frequency for a random hole pattern or for a periodic hole pattern as a function of hole separation and as a function of angular separation.

30. The apparatus according to claim 21 , further comprising:

a fourth unit configured for determining whether pitch frequencies fall into a negative region of the first eigenfunction by using the optimized illumination; and

a fifth unit for separating the pitch frequencies which fall into the negative region into another layer to be imaged with a second exposure, wherein

optimizing illumination for said another layer is performed by the first to third unit.

31. A device manufacturing method comprising the steps of:

(a) providing a substrate that is at least partially covered by a layer of radiation-sensitive material;

(b) providing a projection beam of radiation using an imaging system;

(c) using a pattern on a mask to endow the projection beam with a pattern in its cross-section;

(d) projecting the patterned beam of radiation onto a target portion of the layer of radiation-sensitive material,

wherein in step (b) providing said projection beam includes the steps of:

determining a pitch frequency of a full-chip layer, the pitch frequency indicating how often a pitch occurs in the full-chip layer so as to generate a pitch frequency histogram of the full-chip layer;

equating the pitch frequency histogram to be a first eigenfunction of the sum of coherent systems representation of a transformation cross coefficient; and

solving an integral equation for the first eigenfunction of the transformation cross coefficient so as to define the optimal illumination for the full-chip layer.