Methods and systems for intensity modeling including polarization

1. A non-transitory computer readable medium encoded with computer executable instructions for modeling an intensity profile at a surface resulting from illuminating a mask with an illumination system having an illumination source, including instructions that when executed perform the steps of:

receiving polarization data describing the illumination source;

generating a polarization data model of the illumination source using the polarization data by evaluating a plurality of two-dimensional functions at a plurality of points in a pupil plane associated with the illumination system, wherein the plurality of two-dimensional functions includes: a first function describing polarization of the illumination source along a first axis; a second function describing polarization of the illumination source along a second axis; a third and a fourth function each describing a coupling component of the illumination source between the first and second axes and based at least in part on a phase angle between the first and second axes; and a fifth function describing a non-polarization component based on a degree of polarization;

generating a matrix operator using the polarization data model;

generating a mask function based on the mask; and

convoluting the matrix operator with the mask function to generate intensity profile data representing the intensity profile at the surface.

2. The computer readable medium of claim 1 further including instructions for altering the mask function based on the intensity profile.

3. The computer readable medium of claim 1 wherein the instructions for receiving polarization data include instructions for receiving data analytically describing the polarization of the illumination source.

4. The computer readable medium of claim 1 wherein the instructions for receiving polarization data include instructions for receiving data empirically describing the polarization of the illumination source.

5. The computer readable medium of claim 1 wherein the first and second axes are orthogonal.

6. The computer readable medium of claim 1 wherein the plurality of two-dimensional functions include:

K — XX =cos 2 α( f,g )* DoP ( f,g );

K — YY =sin 2 α( f,g )* DoP ( f,g );

K _cos — XY =sin α( f,g )*cos α( f,g )*cos φ( f,g )* DoP ( f,g );

K _sin — XY =sin α( f,g )*cos α( f,g )*sin φ( f,g )* DoP ( f,g ); and

K _non — pol =1− DoP ( f,g );

where f and g represent pupil variables in a spatial frequency domain, such that coordinates in the pupil plane are defined by an f value and a g value, α(f,g) represents a polarization angle as a first function in the pupil plane; φ(f,g) represents a phase angle between a first and a second dimension polarization as a second function in the pupil plane, and DoP(f,g) represents a degree of polarization as a third function in the pupil plane.

7. The computer readable medium of claim 6 wherein the matrix operator is given by:

K optics=norm( cvr ( J@K — XX )* K pupil X+cvr ( J@K — YY )* K pupil Y+ 2* cvr ( J@K _cos — XY )* Kp cross+2 *cvr ( J@K _sin — XY )* Kp cross — i+cvr 0.5*( J@K _non — pol )* K pupil X +0.5* cvr ( J@K _non — pol )* K pupil Y );

where norm includes a normalization operation; cvr includes a covariant operation; J represents a first pupil function, KpupilX and KpupilY represent second and third pupil functions in a first and second dimension, respectively, Kpcross and Kpcross_i represent third and fourth pupil functions capturing a real and an imaginary coupling between dimensions, respectively, and @ includes a convolution operation.

8. A method of selecting at least one component of an illumination system including an illumination source for illuminating a surface through a mask, the method comprising:

generating, using a processor, a polarization data model of the illumination source by evaluating a plurality of two-dimensional functions at a plurality of points in a pupil plane associated with the illumination system;

modeling, using a processor, an intensity profile at the surface based in part on the polarization data model and a mask function; and

altering, using a processor, at least one of the mask function and the illumination source based on the modeled intensity profile, wherein the plurality of two-dimensional functions include:

K — XX =cos 2 α( f,g )* DoP ( f,g );

K — YY =sin 2 α( f,g )* DoP ( f,g );

K _cos — XY =sin α( f,g )*cos α( f,g )*cos φ( f,g )* DoP ( f,g );

K _sin — XY =sin α( f,g )*cos α( f,g )*sin φ( f,g )* DoP ( f,g ); and

K _non — pol =1− DoP ( f,g );

where f and g represent pupil variables in a spatial frequency domain, such that coordinates in the pupil plane are defined by an f value and a g value, α(f,g) represents a polarization angle as a first function in the pupil plane; φ(f,g) represents a phase angle between a first and a second dimension polarization as a second function in the pupil plane, and DoP(f,g) represents a degree of polarization as a third function in the pupil plane.

9. The method of claim 8 further comprising generating a mask corresponding to the altered mask function.

10. The method of claim 8 further comprising:

receiving polarization data describing the illumination source; and

wherein the act of generating a polarization data model uses the polarization data.

11. The method of claim 8 wherein the act of modeling the intensity profile further comprises:

generating a matrix operator using the polarization data model;

generating a mask function based on the mask; and

convoluting the operator with the mask function to generate a modeled intensity profile.

12. The method of claim 8 wherein the act of altering the mask function includes describing additional features on the mask.

13. The method of claim 8 wherein the act of altering the mask function includes removing description of portions of features from the mask.

14. The method of claim 8 wherein the plurality of two-dimensional functions includes a first function describing polarization of the illumination source along a first axis, a second function describing polarization of the illumination source along a second axis, and a third and a fourth function each describing a coupling component of the illumination source between the first and second axes.

15. A mask for use in an illumination system having an illumination source, to reproduce a predetermined feature at a surface, the mask comprising:

a pattern of at least partially opaque and at least partially transparent features, the pattern selected based on the predetermined feature and based on a modeled intensity profile at the surface generated by the illumination source, the modeled intensity determined by acts including:

receiving polarization data describing the illumination source;

receiving initial mask profile data based on the predetermined feature;

generating a polarization model of the illumination source by evaluating a plurality of two-dimensional functions at a plurality of points in a pupil plane associated with the illumination system, wherein the plurality of two-dimensional functions includes: a first function describing polarization of the illumination source along a first axis; a second function describing polarization of the illumination source along a second axis; a third and a fourth function each describing a coupling component of the illumination source between the first and second axes and based at least in part on a phase angle between the first and second axes; and a fifth function describing a non-polarization component based on a degree of polarization;

generating a matrix operator using the polarization model;

generating a mask function based on the initial mask profile data; and

convoluting the operator with the mask function to generate the intensity profile.

16. An integrated circuit having features constructed, at least in part, by illuminating a surface with an illumination system having an illumination source through a mask, the mask comprising:

a pattern of at least partially opaque and at least partially transparent features, the pattern selected based on the predetermined feature and based on a modeled intensity profile at the surface generated by the illumination source, the modeled intensity determined by acts including:

receiving polarization data describing the illumination source;

receiving initial mask profile data based on the predetermined feature;

generating a polarization model of the illumination source by evaluating a plurality of two-dimensional functions at a plurality of points in a pupil plane associated with the illumination system, wherein the plurality of two-dimensional functions includes: a first function describing polarization of the illumination source along a first axis; a second function describing polarization of the illumination source along a second axis; a third and a fourth function each describing a coupling component of the illumination source between the first and second axes and based at least in part on a phase angle between the first and second axes; and a fifth function describing a non-polarization component based on a degree of polarization;

generating a matrix operator using the polarization model;

generating a mask function based on the initial mask profile data; and

convoluting the operator with the mask function to generate the intensity profile.

17. A method of selecting an illumination source for an illumination system to produce at least one predetermined feature on a surface by illuminating a mask, the method comprising:

Receiving, using a processor, mask data;

modeling, using a processor, the intensity profile generated at the surface when the mask is illuminated using the illumination source, the act of modeling including:

receiving, using a processor, polarization data describing an initial illumination source;

generating, using a processor, a polarization model of the illumination source by evaluating a plurality of two-dimensional functions at a plurality of points in a pupil plane associated with the illumination system;

generating, using a processor, a matrix operator using the polarization model;

generating, using a processor, a mask function based on the mask data; and

convoluting, using a processor, the operator with the mask function to generate an intensity profile; and

selecting, using a processor, a desired polarization for the illumination source based on the intensity profile, wherein the plurality of two-dimensional functions include:

K — XX =cos 2 α( f,g )* DoP ( f,g );

K — YY =sin 2 α( f,g )* DoP ( f,g );

K _cos — XY =sin α( f,g )*cos α( f,g )*cos φ( f,g )* DoP ( f,g );

K _sin — XY =sin α( f,g )*cos α( f,g )*sin φ( f,g )* DoP ( f,g ); and

K _non — pol =1− DoP ( f,g );

where f and g represent pupil variables in a spatial frequency domain, such that coordinates in the pupil plane are defined by an f value and a g value, α(f,g) represents a polarization angle as a first function in the pupil plane; φ(f,g) represents a phase angle between a first and a second dimension polarization as a second function in the pupil plane, and DoP(f,g) represents a degree of polarization as a third function in the pupil plane.

18. The method of claim 17 wherein the plurality of two-dimensional functions includes a first function describing polarization of the illumination source along a first axis, a second function describing polarization of the illumination source along a second axis, and a third and a fourth function each describing a coupling component of the illumination source between the first and second axes.

19. The method of claim 18 wherein the first and second axes are orthogonal.

20. The method of claim 18 wherein the plurality of two-dimensional functions further includes a fifth function describing a non-polarization component.

21. The method of claim 17 wherein the matrix operator is given by:

K optics=norm( cvr ( J@K — XX )* K pupil X+cvr ( J@K — YY )* K pupil Y+ 2* cvr ( J@K _cos — XY )* Kp cross+2 *cvr ( J@K _sin — XY )* Kp cross — i+cvr 0.5*( J@K _non — pol )* K pupil X +0.5* cvr ( J@K _non — pol )* K pupil Y );

where norm includes a normalization operation; cvr includes a covariant operation; J represents a first pupil function, KpupilX and KpupilY represent second and third pupil functions in a first and second dimension, respectively, Kpcross and Kpcross_i represent third and fourth pupil functions capturing a real and an imaginary coupling between dimensions, respectively, and @ includes a convolution operation.